point-set-topology

[crisp lintel]

Certainly the first three are equivalent

and X completely regular should imply those

a problem here is I think that if X is indiscrete then it may be initial for those

yeah because if X is indiscrete the only continuous functions from X to R are constant

and the coarsest topology making constants continuous should just be the indiscrete topology

[median sand]

That's completely regular (I'm using the definition that doesn't include the Hausdorff assumption)

[crisp lintel]

Oh yeah true

I guess completely regular is separating points and closed sets by continuous functions and if the only closed sets are empty and everything then ofc

so I guess suppose the topology is initial. Let x be a point in X and F a closed set not containing x. Find a continuous function f and a closed subset C of the reals with f^{-1}(C)=F

then you should be done

[median sand]

Thanks

[opaque scroll]

I guess there is some work in showing you can find such f.

Since a priori F could be the intersection of such sets

[crisp lintel]

Oh true

I think you can mess with it though

It would be the union though because it's closed no?

[cosmic zodiac]

How might I go about proving "$X$ is normal if and only if it has a compactification $Y$ such that any two disjoint closed sets in $X$ have disjoint closures in $Y$. Any such space $Y$ can be identified with $\beta X$."

[gentle osprey]

supimed

[cosmic zodiac]

It's exercise 6C.3 in Rings of Continuous Functions.

[limber wyvern]

Hi, for proving homemorphism for a function f, is it valid to just proof that $f^{-1}$ is continous. Or do I have to proof that f is continous and biyective?

Thanks in advance

[gentle osprey]

S0S4 - Feel free to ping

[opaque scroll]

f^-1 only makes sense if f is bijective, so you'll need to prove that.

And you do also need f to be continuous, though I guess in most contexts that's more or less given.

[limber wyvern]

Fine, thanks 🙂

[polar storm]

yeah, my only point-set topology application so far was analysis. Hope to see more motivations when I study some geometry and topology in the future.

Oh, actually I was kinda curious about the notion of nets. So I first encountered nets while studying Folland's chapter 4, but then he didn't really use it so far (I'm currently reading chapter 6 L^p spaces). Are nets useful in analysis in the sense of sequential approximation or something similar?

I found it interesting that they provide sequence-like characterizations of compactness, generalize statements that fail for sequences in general topological spaces, and also allow one to characterize topology via net convergence, which does not work if one only uses sequences. But wondering whether this is used and applied a lot in analysis.

[crisp lintel]

You need to use nets when you work topologies that can't be described by sequences basically

and they're useful for the same reason as sequences, often it's easier to prove that a function is continuous using nets rather than messing with open sets

same for proving sets are closed

most of the time for those two applications the use of nets is identical to the use of sequences really and you don't even notice they are nets

[crisp lintel]

The most common examples would be the weak and weak-* topologies in functional analysis I'd say

At least the most common in the areas I'm interested in

[polar storm]

I see. Sounds very interesting. So far, the spaces that I've encountered so far while studying analysis are mostly metrizable lol.

[gaunt linden]

Sounds like that will at least depend on which properties/definition of the function you already have to base the proof on.

[crisp lintel]

yeah

[polar storm]

oh, the FA book that I'm currently reading discusses weak and weak* topologies in the next chapter. Maybe I can see some applications of nets there.

hope to see some

[crisp lintel]

here's like a random example where nets are conceptually nice

though not strictly necessary

since the set of group homomorphisms is just $\bigcap_{s,t\in\Gamma}{\varphi\colon\Gamma\to\mathbb{T}:\varphi(st)=\varphi(s)\varphi(t)}$ which is an intersection of closed sets

[gentle osprey]

Blake

[crisp lintel]

but especially with more complicated examples, the arguments using sets can get convoluted

if you have a ton of intersections of various sets flying around

[gaunt linden]

What is \mathbb{T}?

[crisp lintel]

circle group

[polar storm]

interesting

[crisp lintel]

convergence is also typically easiest to picture in your head using nets

for instance the product topology is just the topology of pointwise convergence

which is more concrete to think about rather than having to think about exactly what the open sets look like

[robust drum]

Nets provide a relatively easy proof of Tychonoff theorem

A set is compact iff “every ultranet converges”

[polar storm]

oh yeah I saw that in Folland's chp 4, which was much easier than that of Munkres using FIP of compactness

[robust drum]

Nets in a product topology converge iff their components do

That being said I’m scared of nets

And I try to avoid them where possible

Because often you’re working with a non sequential topology on something and you just prove something with nets and it’s completely unclear how the proof is different than if you had used sequences

This happens to me a lot in functional analysis

Cause there are many topologies in functional analysis which are just categorically never metrizable

And you do some proof with nets and you’re like huh I literally don’t see how this proof would fail eg for a set that is sequentially closed but not closed

[polar storm]

lol I just started studying more functional analysis after Folland's chp 5 (which was very brief on FA). Hope to see those kind of net applications in the FA book I'm reading tho not sure if this book is advanced enough to cover them lol.

[crisp lintel]

what's wrong with this

this is the good thing about nets

the only time that nets are scary are when you need to really work with the specific index set in a nontrivial way

for instance there's a proof of tychnonoff that goes by diagonalizing a net which sounds painful

[robust drum]

It feels too powerful

[crisp lintel]

It's not really too powerful, the complexity of the topology is captured by the fact that nets are way way more general than sequences

my favorite example is that it's trivial to construct a net of functions from R to R such that each function is zero except on a finite number of points but the net converges pointwise to 1

whereas if you use sequences of course the limit can only be 1 on a countable set

[robust drum]

Hmmm

Interesting

[unreal stratus]

Insert obligatory mention of ultrafilters

[gaunt linden]

Does this notion of convergence with nets correspond to the product topology on R^R?

[unreal stratus]

Wdym?

Oh sorry lol didn't see above

[opaque scroll]

Yeah if you take the poset of all finite subsets of R, and then let the net be the indicator functions then this converges to the constant function in the product topology.

In general if x is in the closure of A then there's a net with values in A converging to x

[crisp lintel]

yeah the product topology is the topology of pointwise net convergence

when you think about a product as a space of functions

[gaunt linden]

OK, so the take-home here is something like the functions with finite support are dense in R^R with the product topology, and every function has a net of finite-support functions that converges to it, but not always a sequence that does.

[cosmic mirage]

is it ever useful to consider a notion like a \kappa-compact topological space for some (probably want regular) cardinal \kappa? This would be a space where open covers admit subcovers of cardinality < \kappa. Equivalently, the space X is \kappa-compact in the sense of category theory in its poset category of open subsets. the usual notion of compactness corresponds to \kappa = aleph_0 here

[crisp lintel]

there's lindelof spaces which is just compact with countable subcovers

past that I don't think it's too useful

we have a huge reason to care about finite things and a pretty big reason to care about countable things

But for higher cardinals it would only be useful if there was something you could do with say sets of continuum cardinality but not larger ones

[cosmic mirage]

yeah makes sense. i gave this some thought as well and couldnt think of anything that uses compactness that could have used non-finite-ness lol

[tender halo]

thats kappa-lindelofness

but since every space is kappa lindelof for some kappa you instead talk about the lindelof number for that space

there are like cardinal inequalities for it and stuff

[limber wyvern]

Hi, Is there a better to proof inyectivity on this homeomorphism than doing the typical: $\frac{x_1}{1-y_1} = \frac{x_2}{1-y_1}$?

This is the function:
$\f: S \backslash {N} \rightarrow \mathbb{R} \ f:(x,y) = ((\frac{x}{1-y},0), x \neq 0$

[gentle osprey]

S0S4 - Feel free to ping

[limber wyvern]

is the homeomorphism for the stereographic proyection

[rancid umbra]

yea, just construct the inverse and show that composing them yields the identity

[limber wyvern]

I already did what I said there, but I'll try that approach because the calculations for proving It on my way were insanely large

[opaque scroll]

I guess there's also the geometric argument: the circle is convex so a line insects it in at most 2 points

[limber wyvern]

But, for doing It this way shouldn't I first proof that the inverse exists and so, proof for biyectivity?

[opaque scroll]

Giving the inverse would be a proof of its existence

[limber wyvern]

Thats a nice point

[rancid umbra]

the geometric picutre is how you construct the inverse

imagine the real line as the x-axis in R^2

[limber wyvern]

Yeah, i have drawed it

[rancid umbra]

the circle minus the north pole is afixed such that the south pole is touching zero

for a point x_0 on the real line, find the point on the line from N to (x_0,0) that intersects the circle

oh i guess this could be a slightly incorrect variant of the picture. may be off by a factor of 2 somewhere

[limber wyvern]

Ill have It mind, I thinked I had to proof that the function is biyective for then using the inverse

[rancid umbra]

either the circle is sitting like how i described it before, shifted upwards, or it hasn't moved and is sitting in R^2 as the usual unit circle

[limber wyvern]

For this exercise is the unit circle

With N=(0,1)

[rancid umbra]

okay, yea, then the same idea applies

[rancid umbra]

its easier now since you can just take the equation of this line

and find where on the line the norm is equal to 1

[limber wyvern]

Thank you very much, tomorrow Ill work on It and maybe send some more questions here😅

@limber wyvern

Thanks for all the ideas

[cosmic mirage]

i assume you can just pick kappa to be the cardinality of the set of open sets? lmao

well yeah that should work

[unreal stratus]

I guess technically any larger cardinality

[cosmic mirage]

oh yeah lol

well depends how you define it apparently

wikipedia says some authors define the lindelof number to be st open covers admit subcovers of <= lindelof numbers

[unreal stratus]

True just I guess it seems most reasonable for kappa compact to mean "for any cover of cardinality < kappa" ...

Oh lol

[cosmic mirage]

yeah i agree thats what it should mean

[unreal stratus]

Sure yeah I didn't think about the Lindelöf stuff

[cosmic mirage]

also i realized i was never actually shown a proof of this fact that compact topological space = compact in its poset category Op(X) of opens lol

[unreal stratus]

It is immediate if you think about what filtered colimits and homs mean

Just a lil weird lol

[cosmic mirage]

yeee i worked this out earlier today

[unreal stratus]

Gg

[cosmic mirage]

was a nice exercise in just throwing stuff around

lol

[unreal stratus]

No re

[cosmic mirage]

the forward direction is like, take a finite subdiagram that also covers X and then it has a cocone so youre done

for the backwards direction you define a filtered diagram of like, n-fold unions of things in your cover and then by the hom thingy one of those must be all of X

in that if none of them are then Hom(X, U_i) = \emptyset and then the colimit is empty despite Hom(X, X) = *

[unreal stratus]

Yee

I guess also like u have the map colim Hom(X, U_i) -> Hom(X, U). This always being an iso is equivalent to saying that you can't have LHS empty and RHS nonempty, which is equivalent to compsctness

[fierce mesa]

Hi:
I have the following question from my exam (worded from memory, so may be slightly inaccurate).

Let: [ F(u(t), t) = -2 + \int_0^t \sin(3u(s)) , ds ] Find the value of $\varepsilon$ such that $F: \mathcal{C}([-\varepsilon, \varepsilon], \mathbb{R}) \to \mathcal{C}([-\varepsilon, \varepsilon], \mathbb{R})$ is a contraction. \

I was told that you can do this question using MVT, but I am not sure how to set it up. Can someone give me any guiders? Thanks. For context, I have another proof that doesn't require MVT, but I am curious as to this method.

[gentle osprey]

Average Maths Student

[fierce mesa]

All I have is that:
[ |F(u, t) - F(v, t)| = |\sin\left(3u\left(t_0\right)\right) | |3u - 3v| ]
But I don't know how to simplify the trig term.

[gentle osprey]

Average Maths Student

[fierce mesa]

My application of MVT here is funny because the fact that I already have sine out the front seems to me that this is a contraction no matter what t or u are, so I probably haven't applied it correctly anyway.

Now that I look at it, the function notation should really be $F_t(u)$ rather than $F(u, t)$.

[gentle osprey]

Average Maths Student

[polar storm]

I think u can apply MVT to the difference |sin(3u(s)) - sin(3v(s))| for each s, which gives 3|cos(c)||u(s) - v(s)| for some c, where |cos(c)| \leq 1. This quantity is easy to deal with to get the upper bound since u can just take it outside of the integral. Also, you can bound the remaining difference |u(s) - v(s)| inside the integral by its sup norm.

Actually, this one is a real analysis question lol

[fierce mesa]

Ohhhh inside the integral? Okay okay

Yes that makes more sense

[polar storm]

yeah

[fierce mesa]

Thank you!

Why I tried to apply it outside of the integral is beyond me 😭

[fierce mesa]

Wait, if we apply MVT with respect to s, wouldn't we have to use chain rule? I.e derivative of sin(3u(t)) is 3u'cos(3u(t))?

Hmmm actually

The setup for MVT would be F(u) - F(v), so it would be with respect to the function (i.e u or v). But this seems a little strange at first glance - taking the derivative with respect to a function?

[polar storm]

oh like treat u(s) and v(s) as two numbers on an interval for each s. What I mean by that is apply the MVT to the function |sin(y) - sin(x)|, which gives |sin(3y) - sin(3x)| \leq 3|x-y| for every pair x,y of numbers. Now substitute u(s) and v(s) inside this inequality

[warped helm]

contraction with respect to what metric on C([-e,e], R)?

is it the (\ell^1) metric? [
d(u,v) = \int_0^\epsilon |u - v| , ds
]

[gentle osprey]

josemom2

[polar storm]

I thought it's sup metric lol

[warped helm]

that would make sense too

[polar storm]

yeah, it gives a nice inequality if I assume sup metric.

[warped helm]

|| you end up needing t < 1/3 right? ||

[polar storm]

||yeah ||

[median sand]

What's the "right" way of completing uniform spaces? Willard/Kelley do it by embedding X into a product of semimetric spaces (metric if X is Hausdorff), Bourbaki does it via filters, but their completion is always Hausdorff (which the Willard/Kelley one isn't).

[crisp lintel]

iirc if you lack hausdorff-ness then the completion isn't unique

I don't know about for general uniform spaces but I'm familiar with the completion for tvs's

cleanest I think is probably using filters

[vagrant sapphire]

Let $(X,d)$ be a metric space. Let $D$ be a dense subset such that $ d(x,y) \leq \max{d(x,z),d(z,y)} \forall x,y,z \in D$. Prove that $X$ is ultrametric \

I suppose we can just take successions $x_n, y_n, z_n \in D$ that converge to any $x,y,z \in X$ and then since the metric is a continuous function the ultrametric inequality holds,i wanted to prove in another way though but couldnt. Any ideas?

[gentle osprey]

Sigfripro

[crisp lintel]

That's basically the way you prove it, any other proof will be a different version of that more or less

you could say like, take the function f(x,y,z)=max(d(x,y),d(z,y))-d(x,y)

then the pre-image of [0,\infty) is a closed subset of X\times X\times X that contains D\times D\times D

so it must be everything by density

but I think sequences is honestly just more clear

[robust drum]

Let $K$ be a compact subset of $\mathbb{R}$, and ${I_\alpha : \alpha \in J}$ a cover by open intervals. How do I show that there is a finite subcover where only consecutive intervals in the cover have intersecting closures?

[gentle osprey]

Luka.s

[robust drum]

I think the condition of intersecting closures isn't really stronger at all, so you can just ask for only consecutive intervals intersecting.

I am trying to do some sort of greedy algorithm proof. At stage k, let x_k be the smallest point in K not yet covered, and let I_k be the interval with the largest right endpoint among those not yet picked that also contain x_k.

But I don't know how to prove this works. I've drawn some pictures that I think it should. But I'm stuck in the technicalities.

[tender halo]

by compactness we can assume J is finite

sort intervals by uhh left endpoint

have a set of intervals youve already taken, when looking at the new interval, check if it adds new points, and if so, remove all the intervals from the set fully covered by the new interval and add it

alternatively, proof by induction, remove leftmost interval from the set, induction on the truncated K, then add it to the answer

[quasi warren]

forgive me for the stupid question but I'm unsure of what a neighborhood of x in X is actually supposed to be

the name doesn't make sense at all for me

neighborhood makes it sound like it's a "close" set but it doesn't contain the Thing in question

why would I consider a subset of U (the neighborhood) and not the set itself

If a subset of U contains x, then U itself will also contain it, right??

[trail charm]

Open sets are nice

[trail charm]

Yes

[quasi warren]

then what's the point of the definition? am I missing something

[quasi warren]

I'm also not very sure what Open means in this context

[trail charm]

What definition of neighborhood are you using

[quasi warren]

I'm coming at it from an analysis approach where it just means the boundary isn't part of a set

[trail charm]

I assume it’s the “a neighborhood of a point x is a set V which contains an open subset U containing x”

[quasi warren]

$U \subset X$ is called a neighborhood of $x \in X$ if there is an open set $V$ with $x \in V \subset U$

ooops

[gentle osprey]

Saarker

[quasi warren]

I'm also not really sure about the definition of a point being on a boundary of a subset

[trail charm]

I think your confusion just comes from not knowing the definitions

You can read the wikipedia page or something for these things, it would probably help

[quasi warren]

I'm reading the definitions like Right now

the textbook is sitting in front of me

I'm a bit confused about where this is going

[trail charm]

In my mind, the motivation for neighborhoods is that studying something globally (ie on the whole space) is hard

Looking at an open neighborhood of a point lets you study something locally, which is less hard

(Also take this with a grain of salt but i have really only seen people talk about open neighborhoods, which is just an open set containing a point)

I suppose one can think of open/closed sets in a topology as being some (very wide) generalization of open/closed intervals in R

[quasi warren]

that's kind of what I was thinking about

(again coming from analysis lol)

[trail charm]

Yeah

I think the intuition for limit points helps you get an idea of everything

Eventually you can define a topology on a set by exactly saying what the open sets are, as long as they satisfy the required axioms

[quasi warren]

so im guessing the advantage is the same as in analysis, it just gives you a general idea of the locality of your point

whereas some points are "reachable" by some sets, some aren't

I guess intervals in this case

[trail charm]

Not quite sure i understand what you mean by this, but i agree with the previous one

Maybe another helpful example of neighborhoods is the idea of (real) manifolds, which (in some sense) generalize R^n

Like the generalization should be something like “this object locally looks like R^n for some n”

And the precise phrasing is “every point has a neighborhood homeomorphic to an open subset of R^n for some n”

Homeomorphic just means like “topologically equivalent”

[quaint spruce]

If you don't understand the point of definitions, see the proofs of the theorems they end up making an entrance in

Or just accept for the time being that you don't get the point and you'll eventually find it out. Although the motivation for the definition of neighbourhood should be very clear if you have done analysis

[indigo nymph]

is it even useful to just read the proof and grasp its idea then move on and read it every now and then in order not to forget it
or is it better not to move on unless you prove each property by yourself even if it takes 2 hours for example

[quaint spruce]

To me "grasp the idea" means that I will be able to reproduce a full proof on my own (maybe not for some of the proofs that are very technical, there you just remember the "key ideas", but this discussion is about quite basic concepts)

[warped helm]

you should understand the theorem’s statement, its hypothesis, and the conclusion it makes first

and its useful to relax the hypothesis to see what breaks

[limber wyvern]

Hi, can a subset of a dense set be also dense?

[wide kayak]

yes

[quasi forum]

why not?

[wide kayak]

even demanding proper subsets

consider $\mathbb{Q} \setminus \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

[gentle osprey]

ManifoldCuriosity

[wide kayak]

both smaller sets are dense in R

[unreal stratus]

Or do R minus 1 point, R minus 2 points, ...

[wide kayak]

yup

[quaint spruce]

or just R...

[limber wyvern]

I mean, I though that for a set to be dense the clausure have to be all the space X

but if I take a subset of that subset

wouldn't it not be dense?

[quasi forum]

no? a dense subset means the closure equals the whole space

the majority of actually useful dense subsets are not equal to the whole space

[limber wyvern]

yeah

writing it make me see it xD

even though a set is a subset of another, if their closure is the same

they are both dense

I see

xd

[quaint spruce]

there are multiple examples showing that this intuition is false, see above

[fierce mesa]

Hi. Can anyone explain what I need to show in this question?

For a space to be separable, there needs to exist a countable and dense subset of this space.

I'm not really sure how you do that for this space, however.

I was reading this answer, but I did not understand the rationale behind it: https://math.stackexchange.com/questions/487319/prove-that-sequence-space-ell-p-mathbb-r-is-separable

Namely, they don't find a countable and dense subset? That is, they instead show l^p is dense in the set of all sequences? Isn't this the wrong way around - we need to find a set inside l^p, not a set that contains l^p?

[cosmic mirage]

im not sure what you mean. what the answers say is that you can construct a set M that is dense inside l^p (R). M will consist of those sequences with finitely many rational numbers and 0s everywhere else. you just need to check M is countable and dense

[fierce mesa]

Ohh, I see. Yeah I was thinking of it the wrong way around

I thought it was proving l^p was dense in M. Thanks.

[cosmic mirage]

ye np

i guess i should note that M is not all sequences

[fierce mesa]

Yeah, it's finite sequences with rational entries right?

[cosmic mirage]

yup

[fierce mesa]

Yeah, I probably should have read that bit

I was confused because I didn't actually know what the dense subset was

[cosmic mirage]

just wanted to clarify since you said l^p is dense in all sequences (there isn't really a notion of density for the set of all sequences, because the l^p norm of an arbitrary sequence need not be finite)

[spare igloo]

I had an idea and wanted to know if it might interest you: I could post weekly point set topology challenges every Monday. Each problem would require a proof, and I would publish a full solution every Sunday

[midnight umbra]

sure, if you've got some good problems banked it could be interesting

[robust drum]

im interested.

[tender halo]

sounds fun, although im curious what level you are aiming at

[spare igloo]

I think I'll adjust the difficulty level based on how quickly the answers are provided

[spare igloo]

I can suggest this as the first challenge; it might be easy, but you'll need to be a little creative.

[dusty talon]

I would assume a lot of people have seen this question before

[midnight umbra]

I mean are you asking me to prove the borsuk-ulam theorem or just cite it

[spare igloo]

I think just citing it is enough, but there are resolutions that don't refer to it

[midnight umbra]

sure

iirc the proof in 1D isnt bad, ||f(x) - f(-x) is continuous and either constantly zero or has a sign change so by IVT has a zero|| should be fine

[spare igloo]

That's fine; I'll try to come up with some harder and more unusual problems for the coming weeks.

[tribal palm]

it’s been too long since last i stopped by this channel 🥺

[sonic crane]

Lol yeah

[tribal palm]

today for funsies ive been looking in detail at the iso of categories of preordered sets and of alexandrov topologies

[tribal palm]

very silly use of my time

i have no regrets

[elder dagger]

whats the answer tho?

[robust drum]

this is super cool

the last leg of the triangle of this is the connection to simplicial complexes

in particular, given any simplicial complex there is a T0 topological space weakly homotopy equivalent to it. The space can be chosen to have a finite underlying set/finite poset corresponding to it if the simplicial complex is a finite simplicial complex

so in principle algebraic topology of simplicial complexes can be studied purely combinatorially

There is a book "Algebraic Topology of Finite Topological Spaces and Applications" by Barmak that seems hella cool to me/that I want to read

May also has some notes on this

I am thinking that when Im a PhD student I will try to do a DRP with some undergrad on this as a good excuse to have the time to read this book

[robust drum]

For a while I was trying to think if there was a way to define homotopy groups using a finite topological space as the model of the sphere as the domain and then in principle it’s super computable for CW complexes

But I think you can’t really do this

In particular I think that there’s no good map crushing the equator for a finite model of the sphere (IE there’s no good comultiplication)

Which is sad

[zealous berry]

breaking news: munkres does not capitalize Cartesian (i.e., he does not respect rene descartes)

[jagged ridge]

based???

[cosmic mirage]

cocartesian coproduct

[snow frigate]

i think it's a sign of respect to not capitalize a mathematician's name if enough time has passed

like abelian groups

it means you've become so important to some theory that your name is inseparable from that of some critical construction

[zealous berry]

no way

id feel more disrespected if i was no longer capitalized

i also know literally no other example

besides boolean ig

[zealous berry]

no one is doing this exercise

[warped helm]

it's instructive

you can use some parts to prove others

[zealous berry]

well, i am certainly not doing this one

[kind marlin]

tbh ive definitely done almost all of those at some point on problem sets

[warped helm]

if you haven't done something like it before, or if it's only your 2nd or 1.5th time doing it, you should do it

[kind marlin]

also sometimes even when a question looks obvious youll hit a step that you dont actually see immediately so its nice to catch that

just pick some of them

[warped helm]

this one is good

[zealous berry]

x/y + w/z = xy^-1 + wz^-1

umm

we want (xz + wy)(yz)^-1 = (xz + wy) y^-1 z^-1

[its easy to show y^-1 z^-1 = (yz)^-1 by showing (yz)(y^-1 z^-1) = id]

= xzy^-1 z^-1 + wy y^-1 z^-1

[wide kayak]

any of them that aren't immediately obvious to you, you should do

[zealous berry]

= xy^-1 + wz^-1

[wide kayak]

eat yer vegetables!

[zealous berry]

= x/y + w/z

just make this slightly more formal and done

[zealous berry]

i dont see merit in proving basic stuff about a structure (vector spaces, groups, R in this case) directly from axioms of said structure and a little bit of thinking

its always busywork

i am also of the philosophy that if i think i could solve a problem in 2 hours then i should just skip it, and i could certainly solve a problem of this caliber in 2 hours

[warped helm]

bruh what

terrible philosophy ill say

[warped helm]

the point is to develop R entirely from ordered field axioms

• completeness

showing that you can do so is worthwhile even though all of it is "obvious"

[zealous berry]

why?

[kind marlin]

not to dogpile but this is kind of a bad habit, doing foundational exercises is really helpful to prime your brain for solving harder ones because you get more exposed to methods

[zealous berry]

this is probably true

[warped helm]

you should restrict that philosophy to "i can definitely solve this problem/i have a game plan on how to solve it" for things you skip

often times in the "i think" you will find that your strategy for proving something breaks down somewhere

but your brain doesnt have enough memory or bandwidth to immediately see that

[zealous berry]

it's too time consuming for me to attempt every single problem in a book that i don't have an immediate solution to (and, honestly, if a problem does have an immediate solution, it's probably just too easy)

[warped helm]

math is time consuming

you certainly dont have to do every problem

[zealous berry]

and whatever holes i accrue from skipping problems i think i can do should be patched in the future, considering i will take courses corresponding to books im reading (or future books will just expound on previous topics and focus on some holes)

[polar storm]

omg, I remember doing this (algebraic law, trichotomy of orders, exponential stuff) from N to R while studying Tao's Analysis I book. That was painful tbh

[polar storm]

I feel ur pain. But no pain, no gain 😔

[tribal palm]

disrespecting descartes, as he SHOULD

have i mentiond how much i despise descartes

[tribal palm]

all of these are instructive but its not #point-set-topology, it’s #groups-rings-fields

[zealous berry]

Although this content is in basically any book with an intro sextion, jts in ch1 of munkres, so I decided to post it here

[zealous berry]

What's wrong with him hahaha

[tribal palm]

guh, dont get me started

[zealous berry]

Ok, you have been stopped

[tribal palm]

this seems like a great read thanks for sharing it 💖

[robust drum]

Np!

[tribal palm]

it was also nice because the prelim chapter validated what i had found as well, meanwhile i think the wikipedia page on alexandrov topologies is rather poorly written

[snow frigate]

sully

[cloud kindle]

I very much do, it gets you used to thinking about arguing about these objects and is the best way to check you know whats happening. One of my annoyances about Hatcher is that I find the book tends to be lacking in these kinds of problems.

Anyway a classic problem of this flavour I suggest you do, a problem I just really enjoy and suggest to anyone learning about rings for the first time, show that you need not assume the additive group of a ring is abelian.

They are back of the envelope arguments that shouldnt take much effort or thought, but it doesnt mean you shouldnt do them. Theyre what keep you grounded while you read, its far too easy to get lost without them

[covert nexus]

yeah these things are very often the grounding of a lot of the more sophisticated structural statements that are the theorems you use more often -- understanding how they work is clearly beneficial even if you can get by surprisingly well without them

more importantly if you're a student taking a class im teaching, those problems will be on the exam

:3

[snow frigate]

i think these ‘basic’ arguments often look very similar to arguments used while proving a more important theorem

[covert nexus]

indeed! and are very much essential if you ever want to follow through those structural arguments with computations

the iso theorems are a lot more useful if you can write down what the isos are

(i wouldn't actually count a statement of the iso theorems as fully correct if you didn't ngl)

[snow frigate]

wow

btw, what’s a ‘discrete topologist?’

[covert nexus]

a joke

because discrete topologies are trivial

bounded holomorphic functions are constant, and perfect abelian groups are all trivial too

[snow frigate]

ah

oh yeah

lol

[quick delta]

Discrete subsets can be very interesting!

[snow frigate]

non-$T_1$ finite topologies are actually kinda cool

[gentle osprey]

lexi

[alpine nest]

non-Hausdorff topologies are never cool

[snow frigate]

don’t tell the algebraic geometers

[covert nexus]

cries in algebraic geometry

[quick delta]

It could be worse :3

[silent garnet]

(remark 1, fig 14)
is this actually a CW complex?
aren't boundaries supposed to contain lower dimensional cells, not just intersect them

[silent garnet]

this is from fomenko fuchs. the defintion of CW there says, that characteristic maps map boundary of a disc into union of lower dimensional cells

[gaunt linden]

No, for a CW-complex it's enough that every point on the boundary maps to somewhere on a lower-dimensional cell. There's no requirement that the whole lower-dimensional cell is hit.

This is a bid of a weirdness of the definition, but whenever it happens, it ought to be possible to homotope the attachment into a complex where the gluing does behave like you expect. So up to homotopy equivalence (which tends to be all people care about when they use CW complexes anyway), you can assume that any complex you're given is glued nicely. Of course that requires a proof, which the text will hopefully supply later.

[gaunt linden]

Note in particular that this says "into" not "onto" -- in other words, the map is not required to be surjective.

[zealous berry]

Why the sully bombs bruh

Let me study how I want to study

[zealous berry]

A trivial one

[cloud kindle]

No ones stopping you from studying how you want to study, but if everyone tells you its a bad idea, maybe pause and consider why. Youre welcome to ignore it, but you should think about it

[zealous berry]

I think this and my previous message are good enough reasons as to why I don't care about trivial, uninteresting problems

[zealous berry]

Idk what's in hatcher, but I guess verifying conseqhences of axioms for more complicated structures has merit

But you can intuit the structure of a group or vsp or R very quickly without doing exercises like these

Whatever, agree to disagree; if it poses a problem in the future then I can always read books differently

[cloud kindle]

Yeah I mean I do disagree and wont be swayed, but if it works for you and youre happy, batter on

[lavish mountain]

mmm yes this problem is too trivial for me guards take it away

[autumn inlet]

Hi I was trying out JCT, and took aid of the proof outline by University of Edinburgh's proof of JCT. Here is the first part of it (Jordan Polygon satisfies JCT) phrased in my words

Is this acceptable?

[cloud kindle]

(Helge Tverberg's proof)

[zealous berry]

i like this proof best

[queen prism]

🦌?

[gaunt linden]

I suppose it is likewise bloody obvious that R^3 minus the homeomorphic image of a sphere is homeomorphic to R^3 minus the unit sphere?

[hexed vault]

i feel its non trivial

[gaunt linden]

It's not even true.

[hexed vault]

wait really ?

damn , i thought it was true

[gaunt linden]

The "Alexander horned sphere" is a famous counterexample -- it is homeomorphic to a sphere, but its exterior region is not simply connected.

[hexed vault]

i should exile myself from topology i feel

[cloud kindle]

If you suspect something might be true in topology with less than 3 assumptions, there exists a bullshit counterexample

[covert nexus]

all good advanced math is indistinguishable from adjective soup \silly

[quick delta]

Topological spaces don’t exist unless they’re CW complexes of finite type
And usually just finite CW complexes

[urban zinc]

Topological spaces are things I can draw, no exceptions

[quick delta]

Draw S^2 x S^1

[worn mortar]

This is 100% true, full stop.

I’ll happily dog pile on this

[covert nexus]

i thouhgt you were against turning everything into combi

[worn mortar]

This is 100% true lmfao

[quick delta]

Just look at my paper

[covert nexus]

i haven't actually seen it lol

[cosmic mirage]

wdym by finite type? not sure if I've heard this terminology before. Is this like finitely many cells in each dimension?

also unfortunately I'm obligated to say that retracts of finite CW complexes are also important :/

[quick delta]

Ok true

[zealous berry]

how much algebra knowledge do i need to be able to read part 2 of munkres (the alg top bits)

[quick delta]

How much do you have?

[zealous berry]

basic group theory

[quick delta]

And can you give me the contents page?

[zealous berry]

sure

hm wait lemme see if the preface specifies

doesnt specify

[quick delta]

You should be fine for all of that

[zealous berry]

ok nice

[quick delta]

You might want a little more algebra experience for 12, but you’re not missing any prereqs in a strict sense

[zealous berry]

i see

ill think about it when the time comes

[robust drum]

This one you can kinda draw tbf

[urban zinc]

not a space, all my spaces are sufficiently low dimensionally embedded :P

[quick delta]

[urban zinc]

flat torus is fake

[quick delta]

I just imagine it as a torus tbh

[robust drum]

I feel like if you can draw a gluing diagram you can draw it

That’s my take

[urban zinc]

tbh just draw a sphere and a circle and two dots on them that you can move independently

[unreal stratus]

You know here's a meme point like

I feel we should say CWable

[ancient narwhal]

notice how this rule has less than 3 assumptions, implying that there's a bullshit counterexample to it

[midnight umbra]

it has 3 assumptions

"if you suspect something might be true" "in topology" "with less than 3 assumptions"

[ancient narwhal]

augh

[midnight umbra]

it's not an if and only if though so there may well still be a bullshit counterexample

[unreal stratus]

Society if pointset topology courses were aimed at end-users instead of being about producing weird counterexamples

[quick delta]

I’m yet to see a use for the distinction between connected and path connected outside an intro topology course 🙃

[unreal stratus]

Lol

I mean I think this is reasonable because both notions r useful so nice to know they're different but yes

To me it is amusing as in htpy theory you almost exclusively use path connectedness but for algebraic geometry you would only ever use connectedness (or some other variant besides path connectedness)

[quick delta]

Any topological space that isn’t metrisable and locally path connected is pathological:3

[unreal stratus]

Lol

My cantor set

[quick delta]

Hmmmm
Locally path connected or totally disconnected

Profinite stuff is useful i admit

And I kinda wanna think about the boundary of F_2

[unreal stratus]

Field with two elements

[quick delta]

[cosmic mirage]

me when i adjoin an element to F_1

[ancient narwhal]

field with only elements

[snow frigate]

topology on the free group

[zealous berry]

are coarser and finer topologies relevant to weak topologies?

[wide kayak]

I think weak topologies are generally the coarsest topology that make some desired functions continuous

[zealous berry]

wdym by "some desired functions"

[wide kayak]

for example, if X is a vector space over k and X* its dual space, there's a pairing map X x X* -> k, sending (v,f) to f(v).

Coming from that pairing, for any fixed f in X*, there's a map <-,f>: X -> k, given by sending v in X to f(v).

Now the weak topology for X is the coarsest topology for which all those maps <-,f> are continuous, as f ranges over all of X*

there's also a dual weak* topology on X*, the coarsest topology on X* making all the maps <v,-> continuous as v ranges over X

[zealous berry]

<f, v> is a duality pairing in this case?

[wide kayak]

yeah, vector and covector pairing

[zealous berry]

fixing the covector f, you get a scalar valued function on X

[wide kayak]

right

[zealous berry]

the weak topology is the coarsest topology such that all maps v |-> f(v) are cont

for fixed f in X*

[wide kayak]

there's just one map per f, so f ranges

[zealous berry]

and weak* is analogous but you fix the vector and let the covector vary

[wide kayak]

yup

[zealous berry]

i see

yeah i guess i understand the def but id need to see examples (in the future) to actually understand haha

[wide kayak]

there's a similar idea with the product topology on X x Y

[zealous berry]

like i wouldnt understand what coarsest means here, or a specific topological space (i guess funciton space) where this is nice

wait

is X* topological dual?

oops i interpreted it as a vector space dual

[quasi forum]

its continuous dual generally

[snow frigate]

yeagh

[quasi forum]

or topological whatever you want to call it

theres probably a few names

[wide kayak]

the product topology on X x Y is the coarsest topology in which the projection maps pi_X and pi_Y are continuous

[gentle osprey]

lexi

lexi

[zealous berry]

omg texit online

[snow frigate]

aka zariski is the coarsest topology making polynomials continuous

omg

[wide kayak]

great example!

[zealous berry]

oh nice

ill have to do that soon

(product topology)

[quasi forum]

this is the correct definition for arbitrary products btw

[zealous berry]

pi_X and pi_Y are very fun

[zealous berry]

why not the categorical one

[quasi forum]

if you try to do the "obvious thing" where you take products of open sets you get a topology that is way too big

so it loses an assload of good properties

[snow frigate]

this is equivalent (and it's easy to see why)

[zealous berry]

ohh

on topological spaces i see

[wide kayak]

I think it can be phrased categorically, with the coarsest topology being an initial object in some poset category of topologies on X

[zealous berry]

so the product topology on X x Y is not the cartesian product of T_x and T_y?

[quasi forum]

for finite products it is

for infinite products no

[snow frigate]

weak/strong topology translates very well to categoric definitions

better than other constructions

[zealous berry]

i see

[snow frigate]

i saw this on mathoverflow, it blew my mind

[warped helm]

(pedantic note, even in that case it wouldnt be the cartesian product since that would give you things like (U,V) but we want U x V )

[wide kayak]

you'll learn the difference in Munkres (the one you said is the box topology)

[quasi forum]

yeah ill let it slide in this case because i think its clear enough from context

[zealous berry]

amazing

munkres doesnt touch weak topologies i guess?

[warped helm]

idts

[polar storm]

yeah

[zealous berry]

oh literally next page he specifies

[wide kayak]

he might, I don't think my class covered that though

[quasi forum]

not really, but product topology gives a decent idea on the basics of how this construction works

[snow frigate]

i think his book is more grounded, strong/weak topology feels more categoryish to me

[quasi forum]

you do the same thing except instead of taking the collection of functions to elements of the dual, you take them to be all the coordinate/projection mappings

[queen prism]

well there's "weak" and then there's "weaker"

[snow frigate]

he will use "strongest/weakest topology such that xyz maps are continuous" actually

when defining quotients at least

quotients are a great example of strong topologies

[zealous berry]

alright

i will complete chapter 2 and then revisit this convo

[snow frigate]

quotients are goated

[quasi forum]

anyway you will deal with weak/weak-* topology in functional analysis

[zealous berry]

awesome!

[quasi forum]

the proof that weak/weak-* topology is hausdorff requires hahn-banach so yeah good luck doing that without FA

[zealous berry]

mfw everything i'm interested in is somehow in functional analysis 😂

[snow frigate]

for me quotients are when topology got to "yay we can play around with shapes now"

and when the advantages of working in the generality of topological spaces over metric spaces became clear

[quasi forum]

well unless the vector spaces are finite dimensional since then they all coincide

but thats not a very interesting case

[snow frigate]

i remember when i would avoid working in topologies like the plague

[ancient narwhal]

that sounds pretty reasonable i wouldn't want to work in a topology like the plague

i love being in new servers where people aren't sick of me making this format of joke

[urban zinc]

the "weak topology" is a specific topology related to dual spaces in linear algebra

as other people said, it becomes important later in functional analysis

the weak topology is indeed weaker in general than other natural choices of topology like the norm topology (if you choose the right definition of "weaker"...)

[wide kayak]

I was a bit sloppy with the example I mentioned, should have said X is a topological vector space and X* is the continuous dual

[zealous berry]

is there some reason munkres avoids saying something is an element of a collection of sets

he always says "containing" (e.g., an open set is contained in a topology, rather than is an element of), and whenever it's probably easier to just use the \in symbol to show a set is in a collection of sets, he refrains

[covert nexus]

not a good one as far as im aware

[cosmic mirage]

I think it's easy for people learning this stuff to get scared of sets of sets

(even though a set theorist might say the only things a set may contain are sets)

[zealous berry]

it's more confusing to me when he avoids "set of sets" and keeps treating it as a "collection" without the usual set properties

[cosmic mirage]

i have yet to see a proof that there exists a good topology book so

[zealous berry]

munkres is good so far (although im in the very early parts of the book)

[urban zinc]

it's probably easier to keep track of that way

[zealous berry]

i just translate it to set of sets

also makes it easier to realize a topology on X is just a subset of P(X)

[urban zinc]

really there's three relevant levels here: points, sets of points, and collections of sets of points

having a name for each level is nice

[zealous berry]

i see

[urban zinc]

point-set topology really is about points, sets, and topologies

[cosmic mirage]

aka an element of P(P(X))

and of course one can consider the set of all topologies on a space. Which is an element of P(P(P(X)))

[zealous berry]

oh i never considered this

aweosme

[zealous berry]

any Applications of this.

what is the cardinality of all topologies on R

,w cardinality of all topologies on R

[gentle osprey]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by altanis_

[zealous berry]

noob

is there some alternative formulation of a topology perhaps using closed sets?

[unreal stratus]

yes, since closed sets are precisely the complements of opens, you can just take complements everywhere in the definition

[unreal stratus]

i would guess it is the same cardinality as that of P(P(R))

[zealous berry]

sounds right

[zealous berry]

if you had a topology defined in terms of open sets, the analogous topology that is the same (just formulated in terms of closed sets) is just the complement of each element in the topology of open sets then?

[unreal stratus]

sure except a topology is by definition about open sets so you shouldn't call that a topology aha but yes

[cosmic mirage]

so aleph_3

[unreal stratus]

Lol

This annoys me when people say uncountable to mean at least cardinality of R

[cosmic mirage]

I believe in the generalized continuum hypothesis

actually nah no I dont

[zealous berry]

couldnt a textbook theoretically define a topology in terms of closed sets though

[unreal stratus]

sure but then it's bad calling it a topology but yes you can define topologies using closed sets

[zealous berry]

is there any merit in doing it that way

[cosmic mirage]

topology and groupoids starts by defining a topology using neighborhoods

kind of cursed

[unreal stratus]

it's equivalent and like sometimes more convenient

[cosmic mirage]

I mean they are equivalent notions. When doing topology you should understand that the data of open sets, closed sets, and neighborhoods uniquely determine each other

and that you can and should switch between these perspectives when doing so is convenient

[unreal stratus]

for example there are the co-blah topologies where you DECLARE that the open sets are precisely those whose complements have cardinality blah

Like cocountable or cofinite topologies

But it'd be easiest just to say the closed sets are finite

lol

[zealous berry]

hmm

[unreal stratus]

though saying neihgbourhood is funny cause like

[zealous berry]

that reminds me of generating a topology by a basis

[unreal stratus]

literally just an open set containing the point

lol

[cosmic mirage]

Also in AG it's useful to define topological spaces in terms of open sets, ie Zariski topology

Wait I take that to be open neighborhood

[zealous berry]

ok great

[cosmic mirage]

and neighborhood to mean a set containing an open that contains the point

[unreal stratus]

No what you can do is define the Zariski topos algebraically and show abstractly that this can be represented as sheaves on a topological space

[cosmic mirage]

one of the statements of all time

[zealous berry]

https://tenor.com/view/super-mario-bros-wonder-talking-flower-template-mario-wonder-meme-gif-7964213445666430726

[opaque scroll]

Beth_3 not getting enough love

[unreal stratus]

I agree

Beth numbers are fun

[opaque scroll]

ב

[zealous berry]

now munkres treats collections of sets as sets!

[novel acorn]

??????

[unreal stratus]

What's the issue lol

[novel acorn]

Yeah lol idk what the problem is

[unreal stratus]

Collection in this context just means set

[zealous berry]

^

[unreal stratus]

Well saying A is contained in X can mean either that A is an element or a subset

[novel acorn]

It feels like he's conflating

The word set and collection tbh

And like

There are genuinely

No set theoretic issues that arise

In the stuff Munkres is doing lmfao

[unreal stratus]

Iirc he just says "collection of sets" instead of a "set of sets"

Like it isn't meant to be something more nebulous

[urban zinc]

also one in terms of neighborhoods, also one in terms of closures

[cosmic mirage]

also huh i never realized you can do this for any (infinite) cardinal, but thats nice

like a topology where kappa-small things are closed

[zealous berry]

is it acceptable to say, for topological space $(X, \mathcal{T})$, that some $U \subseteq X$ is open \emph{under} $\mathcal{T}$

in the event im talking about multiple topologies

[warped helm]

not used

[lilac musk]

I've seen open in, not under

[zealous berry]

wdym

[zealous berry]

i see

[warped helm]

its not wording thats used anywhere

[zealous berry]

oh

[warped helm]

to specify this data you would say “U is open in X with the Tau topology” since theyre usually given names

[zealous berry]

that's long though

ok so is U subseteq is open with Tau?

in sounds good though as knightwatch said

[warped helm]

which is why its shortened to “U is an open subset of X” when the context is clear

[warped helm]

you would not say “open in Tau”

you say open in the space X

the elements of Tau are what the open sets are

[polar storm]

I'd say something like "$U$ is open in $(X,\mathcal{T})$"

[gentle osprey]

Euiseok (Class of 1929 + 100)

[queen prism]

open with respect to T

[zealous berry]

fine

wrt is good

i changed my mind

i am using "under"

[zealous berry]

the problem with this is that i dont want to keep writing out (X, Tau)

[zealous berry]

and this because i dont want to write "with respect to"

[polar storm]

Fair enough.

[zealous berry]

i will Revolutionize pointset topology

[polar storm]

I don't think this would cause any confusion, so I think it's totally fine to use that

[polar storm]

🤯

[polar storm]

I just use "in (X,T)" when I write my point-set topology notes.

[queen prism]

then just say U \in \mathcal{T}

I personally would say [ U \overset{\text{open}}{\subseteq} (X, T) ]

[gentle osprey]

nBladeoid

[zealous berry]

nah i want to emphasize it's open to help me make the link between elements of a topology and open sets

[zealous berry]

[lavish mountain]

i guess if you're ridiculously strapped for time, you cld say U \subseteq X is T-open

[urban zinc]

just say U \in T

[radiant stone]

a schizo way to say an open set be like let (U\hookrightarrow X) be an open immersion

[novel acorn]

Okay

[iron bolt]

in differential geometry, the mapping class group of a manifold is often defined as the group of self-homeo/diffeomorphisms of the manifold, modulo path components, i.e. the quotient group where each path component gets sent to a single point

this makes me wonder: is this quotient totally path-disconnected? generally when you have a topological group you can always take its quotient modulo path components - is every path component in that quotient a singleton?

in general that isn't true, it's not too hard to construct spaces with multiple path components whose quotient modulo path components is indiscrete

but I don't know how you could modify such a counterexample in such a way that it becomes a topological group

[craggy sage]

Let X be a topological space such that every bounded subset A \subset X is closed. (There still can be closed sets which are not bounded.) Can we classify such X?

[cosmic mirage]

What do you mean by bounded here?

I don't know of a notion of a bounded set in a general space that isn't, say, a normed vector space

[crisp lintel]

In a metric space at least, this implies every set is closed (hence the space is discrete). Indeed, let $C\subseteq X$ be an arbitrary subset. Suppose $(x_n)$ is a sequence in $C$ with $x_n\to x\in X$. Convergent sequences are bounded, so ${x_n}$ is closed. However, $x$ belongs to the closure of the sequence, so in fact $x=x_m$ for some $m$, implying $x\in C$.

[gentle osprey]

Blake

[crisp lintel]

I guess you could consider a topological space that is also equipped with a bornology https://en.wikipedia.org/wiki/Bornology

if you assume that (1) every compact set is bounded and (2) the space is first countable, the same argument as above implies the space is discrete

[polar storm]

This is from Munkres' book. I tried to generalize it to arbitrary products just for fun, and I’m wondering whether my proof is right or contains an error I haven’t noticed.

[gentle osprey]

Euiseok (Class of 1929 + 200)

Euiseok (Class of 1929 + 200)

[midnight umbra]

looks good to me

[polar storm]

Thank u

[spare igloo]

Here is the second challenge.I think that's at least a nice problem

[hidden spade]

why do we need to assume the hypothesis of first-countability?

[warped helm]

because there are counter examples

see munkres for examples

[polar storm]

What is the standard name for the version of Urysohn’s lemma for LCH spaces that is used to prove the Riesz representation theorem? Both papa rudin and Folland calls it Uryshohn's lemma, and I'm just wondering if this is the standard name cuz we also have Urysohn’s lemma for normal spaces (tho the context is obvious, so I don't think I'd be confused).

[zealous berry]

is there some symbol that denotes the topology a basis generates?

like perhaps $\mathcal{T}(\mathcal{B})$ or something lol

[polar storm]

hmm I don’t think I’ve ever seen one. At least, Munkres doesn’t have a notation for that.

[midnight umbra]

I haven't seen it but you can always introduce it yourself in anything you're writing

[polar storm]

lmao I think I saw that exercise in Conway's complex analysis I book in metric topology chapter.

[zealous berry]

what's the point of the K-topology

[polar storm]

counterexample. Tbh, I've never seen it being used other than that lol.

also, the lower-limit topology

[zealous berry]

how should i visualize the lower limit topology

[polar storm]

Something like [ )))))))))))))))?

[zealous berry]

all open sets wrt that topology are unions of form [x, y)

idrk what this visually means though ig?

well ive already learned, under the standard topology of R, a set U is open if, for each x in U, some open neighborhood in U contains x

[cosmic mirage]

how are you defining an open neighborhood

[zealous berry]

(x-e,x+e)

[cosmic mirage]

ah yeah sure

[zealous berry]

that's what i learned in like analysis

for the specific case of R with the open interval topology

[cosmic mirage]

My definition of open neighborhood of x is an open set that contains x

[zealous berry]

so idk how to imagine open sets in R endowed with the lower limit topology with this intuition

i guess like the buffer only needs to exist on the rhs

[quasi forum]

the problem is that in arbitrary topologies "open neighbourhood" refers to elements of your base

generally speaking at least

[zealous berry]

yeah that's true

do i have to throw out my intuition of neighborhoods in R with the standard topology then?

[quasi forum]

essentially

[zealous berry]

awesome

[quasi forum]

in almost every analysis context you give R^n the standard/euclidean topology but since this is point set topology your open sets are gonna look different

if you give it the discrete topology then every set is open

[zealous berry]

well munkres hasnt used it yet he just defined them for some reason

[cosmic mirage]

yeah I feel like the only topologies on R that I understand are the standard one, the discrete one, and the silly one with no nontrivial open sets

[cosmic mirage]

like at least intuitively / visually

[zealous berry]

and for exercise i have to show the k-topology and lower limit topology are incomparable

[quasi forum]

mm ok

[polar storm]

More importantly, the lower limit topology is not metrizable.

[quasi forum]

i mean the point of those topologies is for instructive purposes

just to get you used to topologies that arent metric or whatever

[balmy briar]

Continuous functions would be the right continuous functions in R

[zealous berry]

interesting

[zealous berry]

right continuous means?

[quasi forum]

in practice yeah i havent seen lower limit topology anywhere outside of a point set topology textbook

$\lim_{x \to a^+} f(x) = f(a)$

[gentle osprey]

KraySovetov

[balmy briar]

limit from the right exists

and equals the function at that point

[zealous berry]

oh okay

alright well i wont think too hard about it until the time comes

[quasi forum]

certainly you should at least get used to the fact that the same space can be given different topologies that behave wildly differently

with the continuous dual you already have 3 topologies to worry about in functional analysis

the one from the operator norm, the weak-* topology and the weak topology

theres also even more of them in operator theory stuff but i dont know how those work

[cosmic mirage]

a fun thing to think about is how many non-homeomorphic topologies you can give a finite set with n elements

this has no known closed form solution but it's fun to think about

but you can find some easy bounds that grow very quickly

[quick delta]

There is 1, the discrete topology
:3
non-hausdorff topologies are a lie

[cosmic mirage]

a lie what? Group?

[quick delta]

😠

[cosmic mirage]

geometry named algebraic:

[kind marlin]

this kind of reminds me of the union-closed sets conjecture

i guess this should be easier

[cosmic mirage]

what is that conjecture?

[kind marlin]

https://en.wikipedia.org/wiki/Union-closed_sets_conjecture

for every finite union-closed family of sets, at least one element belongs to at least half of the sets

and u can reduce the statement to just looking at finite families of finite sets

[cosmic mirage]

huh. i am surprised this is open lol

[kind marlin]

right??

it's annoyingly easy to state

[cosmic mirage]

this is trivial. look at it

[kind marlin]

these results are so evil 😭 the unpublished preprint takes the cake though... how close to 0 is that epsilon

[iron bolt]

huh. I somehow thought I had heard at some point that the conjecture got proven

but maybe I'm just misremembering those 2022 results

[shell star]

Let K be a finite extension of Qp, or for convenience pretend K=Qp. Im wondering about properties of ℙᵐ(K) with the topology from the norm on K. you can show by covering ℙᵐ(ℚₚ) with [Zp,Zp,1,Zp,...,Zp] that this space is compact, it's also clearly totally disconnected with no isolated points. Using the Bruhat-Tits tree you can construct an explicit isomorphism between ℙ¹(K) and the cantor set, is there an isomorphism for m>1? It satisfies already the requirements to be cantor except metrizability which feels plausible. It looks like you can make a metric from that cover by products of the valuation ring as i couldn't find a contradiction in that at least for m=2. Thats why im asking in this channel since the question becomes what kinds of conditions do you want to take surjective map and get a metric on the image that generates the topology on the image

something like locally an isomorphism + finite fibers maybe. [Answer: Locally an iso is enough in most cases as paracompact + hausdorff + locally metrizable <=> metrizable]

ah i think this specific case should be doable with Urysohn you can get regularity by taking P^m=P^{m-1}uA^m and organising such that your point is in the affine bit then taking an epsilon neighbourhood, wait in fact compact hausdorff second countable is enough

[spare igloo]

That's entirely possible, but I didn't find it there

[polar storm]

I see. Btw, I didn't know that this notion is called "chain-connectedness." I found this result quite interesting.

[dusk tree]

What's a quick and easy proof to show that compact subsets of the irrational numbers have empty interior?

[robust drum]

If a set is compact in the irrationals it’s compact in the reals and therefore closed in the reals. A subset of the irrationals that is closed with respect to the topology on the reals is discrete as a subspace of the reals. A discrete compact set is finite. An interval of irrationals contains infinitely many irrationals so no finite set can have an interior as a subspace of Q.

I think something like this should work

[opaque scroll]

If it has non-empty interior it contains all the irrationals in an open interval (a, b). Shrinking the interval we may assume a and b are rational.

Then (a, b) is clopen and you can just use a typical infinite cover like
(a + 1/n, b)

[dusk tree]

Thank you guys

[alpine nest]

"at most 50 sets", I love this

[quick delta]

A subset of the irrationals that is closed with respect to the topology on the reals is discrete as a subspace of the reals.
This is false right, consider {pi, pi + 1/n | n \in \N}?

[quick delta]

(Bap)

[robust drum]

Good point. Idk why I thought it was.

[midnight umbra]

let I be the set of irrationals. suppose for a contradiction a compact set K had an interior point x. then, there's some open neighbourhood U of x in R such that U n I is a subset of K. by the density of the rationals, there exists a rational number q in U. we can then find a sequence of irrationals in U n I converging to q. this has no subsequence that converges in I

[zealous berry]

what about U is Tau-open

[gaunt linden]

Whereas saying U \in \tau would be technically correct, but feels like a bit of an abstraction break.

[warped helm]

i guess you could write this, in practice ive never seen this

usually just something along the lines of saying U is open in X ; its not often the case that you’re rapidly switching between many different topologies to where you have to write something like this to keep track of it

or whether you care about the specific topology to begin with and not just the fact that X is a topological space with open sets

[crisp lintel]

The irrational numbers have empty interior so every subset does as well

Don't know why people are overcomplicating this

For a proof of the first fact, all you need is the fact that every open interval contains a rational number

[gaunt linden]

I understood it as being the interior relative to R\Q that was empty.

[crisp lintel]

Ah ok, that's more involved then

[cosmic mirage]

@zealous berry is it possible to like

say let X be this set and X_1, X_2 be these different topological space structures on X

well don't phrase it like that, I just woke up, but the point is you can just add a dummy index

[zealous berry]

and then say U \subseteq X_1 is open, meaning U is T_1-open?

[cosmic mirage]

yeah I feel like that is clear

[zealous berry]

sounds good as well

[cosmic mirage]

like this is an abuse of notation but it's no worse than what is already standard to do

whereas saying T_1-open isn't standard so can cause a bit of friction when reading it

Well if I read T_1-open I would think I was lacking knowledge of some standard definition about how open sets relate to the T_1 separation condition, lol

[zealous berry]

how so

[cosmic mirage]

Well if X_1 is a topological space, X_1 isn't the set underlying the topological space. Depending on how you set things up X_1 is probably a pair, whose first element is the set you're talking about

But it's standard to abbreviate U \subseteq first element of the pair X to just U \subseteq X, or more generally to pass back and forth between X being the set vs the topological space structure on the set

this is the same thing as like, let g \in G be an element of a group G, or X \in C be an object of a category C

[zealous berry]

Just overload \in so it's defined on topological spaces

[sly geyser]

Is a subset of a relatively compact set relatively compact?

Hmm so any closed subset of a compact space is compact

Now let me look at the proof of this

[zealous berry]

yea

[warped helm]

A subset C implies clA subset clC and clC would be compact in this case, so clA is closed and so compact

[sly geyser]

I've noticed that the functional analysis text I'm reading assumes knowledge of topology in cases like this (instead of proving that (boundedness <-> relative compactness for all subsets) <-> finite-dimensionality <this is for normed spaces>, they proved that (unit ball is relatively compact) <-> space is finite-dimensional)

These vector spaces are over R or C

When reading this text I've always been curious about what fields other than R or C could be used to be honest

Though perhaps I should be talking about this in advanced-analysis

[zealous berry]

i dont understand the point of a subbasis

from what i understand, it's just a cover of a set such that the set of all finite intersections forms a basis

wait yeah actually what is the diff between a subbasis and cover

[polar storm]

A subbasis is basically a collection that is even simpler than a basis, but still generates a topology

[zealous berry]

is every cover of X a subbasis

[polar storm]

yeah, but taking some arbitrary cover of X won't give u the basis that generates the topology u want.

[cosmic mirage]

Yeah the topology will often be finer

[zealous berry]

sure

[cosmic mirage]

@zealous berry the point of a subbasis is that the construction gives you a recipe for the minimal topology that has those sets be open

[warped helm]

the product topology is an example where you want a description in terms of a subbasis to be able to work with it

[polar storm]

basis and subbasis are useful cuz they are much simpler/smaller collection than that of topology, but they have enough information that many topological arguments can be simplified/reduced to just proving it for basis/subbasis which are much easier to work with.

[unreal stratus]

One distinction to underline here is that there is the notion of a subbasis for \tau where \tau is a given topology. Saying something is a "subbasis" is not a particulary interesting notion because basically any collection is a subbasis

[zealous berry]

yeah thats what i understood about a basis

like you can show a topology is finer than another by showing basis elements can be locally refined

[unreal stratus]

Like the point is given a basis or a subbasis you can generate a topology T. Then given your T you can ask whether a given (sub)basis generates that T, i.e. they are (sub)bases for T

[cosmic mirage]

yeah to specify a subbasis you don't need any of the weird axioms open sets have to satisfy, which is nice

[zealous berry]

i see

i just dont understand the placement of this definition cuz its not used anywhere yet

[cosmic mirage]

which is nice because for example, it's useful to declare certain sets to be open (for example when you want to force certain maps to be continuous, such as projections onto each coordinate in the case of product spaces)

[zealous berry]

sure

ahh i see

[quasi forum]

in practice this is how i came to appreciate the concept

[zealous berry]

a basis requires an extra axiom besides covering

[quasi forum]

you need topologies to be closed under finite intersections and arbitrary unions

[zealous berry]

so a subbasis is weaker in the sense that, when generating a topology from the subbasis, you simply just declare certain sets to be open without the "intersection must contain a basis element" requirement for bases

[unreal stratus]

I would phrase it as a basis being a convenient subbasis (becuase the topology they generate is more easily described in terms if your subbasis is a basis)

[cosmic mirage]

yeah me too, it was never really explained to me or never explained well which is a shame, bc it's so elegant

[quasi forum]

once you think of it that way its obvious why the construction looks the way it does

[unreal stratus]

I would say that bases and subbases feel a bit overly emphasised in intro topology and are often a bit confusing initially, like once you have some examples it'll be more clear why they're useful

[cosmic mirage]

trillions must close a subbasis under the axioms of a topology

[zealous berry]

a subbasis generates a basis generates a topology right

[cosmic mirage]

yup

[quasi forum]

yeah

[polar storm]

As a very basic example where basis is good, if u want to prove that a function f is continuous, u only need to check it for inverse image of basis elements, instead of arbitrary open sets

[zealous berry]

idk the def of continuity with open sets yet

or well like i kinda do

[cosmic mirage]

preimage of open is open

[polar storm]

also this works for subbasis

[zealous berry]

a function is continuous if the inverse image of an open set is open

but i havent really internalized that yet cuz i havent gotten there yet so

[cosmic mirage]

and you can check continuity on a subbasis which is useful bc that's how preimages work

[unreal stratus]

Ah yeah here's the important question

have you done any analysis Altanis

[quasi forum]

its pretty easily checked to be equivalent to epsilon-delta for metric spaces

[zealous berry]

yeah

[unreal stratus]

nice

[quasi forum]

there is also an alternative version where its like

f is continuous if for every open V in Y there is an open U in X such that f(U) \subseteq V

which is also not hard to show is equivalent

[zealous berry]

awesome

[quasi forum]

this looks a lot more like the standard epsilon-delta definition

and it pretty much is

[cosmic mirage]

wait why can't I just take U to be empty

[quasi forum]

you need to specify a neighbourhood around a point iirc

[cosmic mirage]

ah sure

[quasi forum]

if you want it to be fully correct its for all x \in X where V is a neighbourhood of f(x), there exists U a neighbourhood of x such that f(U) \subseteq V

then its correct

[zealous berry]

i made an analogy between topological spaces and vector spaces in the sense that a basis for a topology (resp. vector space) generates every element in a topological space (resp. vector space) under arbitrary union (resp. linear combination)

[quasi forum]

otherwise yeah it fails because of stupid reasons

[zealous berry]

and the difference is that the expression of a vector in terms of a basis is unique

whereas a set in terms of unions of basis elements aren't

[cosmic mirage]

yeah these are both examples of free constructions

[zealous berry]

i hvent gotten to the part about free constructions in aluffi

cuz i paused it for the time being

[cosmic mirage]

nah but this idea of generation pops up literally everywhere

algebra, linear algebra, topology, measure theory just to name a few

[kind marlin]

pi-lambda theorem <3

[zealous berry]

yeah

[polar storm]

You can try proving this for real-valued functions on R as an exercise.

[zealous berry]

measure theory?

oh ig sigma algebras

[quasi forum]

yeah

[zealous berry]

and borel algebras

[cosmic mirage]

exactly

[polar storm]

it's a special case

[quasi forum]

you check a property on a collection of sets which generates the sigma algebra

then you use pi-lambda or monotone class to get it for all sets in the sigma algebra

[zealous berry]

awesome

[quasi forum]

standard technique

its used in the proof of fubini for example

you check on boxes because there its easy, then you monotone class/pi-lambda to get it for all measurable sets

[zealous berry]

i have done this as an exercise already

(for f:R->R)

but i didnt like the inverse image for open set proof cuz i had to assume the domain was R and not a subset of it

i think i tried f:X->R for X subseteq R and it was bad

[alpine nest]

Good times

[cosmic mirage]

goated theorem

[quasi forum]

yeah its very helpful

[zealous berry]

so if i want to declare U, V to be open subsets of X, i can just generate a topology from the subbasis {U, V} (assuming U and V cover X)?

[cosmic mirage]

I remember working on a problem at like 2am a few years ago and finally being able to solve it due to monotone class. Still chasing that high

[quasi forum]

sure, the topology will be very boring (and you can just describe it explicitly) but you can do this

[alpine nest]

In general you often don't even need those, you just show that "property X holds for a certain family of sets A" and "the family of sets with property X is a sigma-algebra", from which you immediately deduce that the family of sets with property X contains the entire sigma algebra generated by A

[quasi forum]

yeah pi-lambda and monotone class are basically just shortcut versions of that idea

because in the proof of those theorems you show that the appropriate collection, whatever it is, contains the sigma algebra with all borel/measurable sets (whatever your sigma-algebra happens to be)

[kind marlin]

yeah you can basically make any collection of sets a subbasis, and then you're just asking "whats the minimal topology containing these sets"

[zealous berry]

@cosmic mirage from what i understand about free constructions, you have a generative set S and F(S) the free structure generated by S. it's free if, for any other structure A of the same "type," any f: S -> A can be identified with a unique homomorphism phi: F(S) -> A

[cosmic mirage]

yee

[zealous berry]

if you build a free group from {a, b}, and you have x,y in G (G being a group i guess in this case), there's a unique way to map the free group into G by sending a->x, b->y

[cosmic mirage]

exactly

[zealous berry]

great!

[cosmic mirage]

something like this should also work for like. Fix a set X, and then consider topologies generated by a collection of subsets of X

I haven't checked the details but if this was false that would be absurd

[kind marlin]

if youve gotten to product topologies thats a rlly common place to see this, we basically just enforce wanting projection maps to be continuous, and then determine what the resulting minimal induced topology is

[zealous berry]

yeah this is a good picture

[cosmic mirage]

(and as a special case of this, if you replace the word group with k-module for some field k, you get back the concept of a basis)

[zealous berry]

i see yeah

k-modules are vector spaces over k no?

[quasi forum]

weak, weak-* and product topologies i think are the best instance of this idea

at least in a concrete setting

[cosmic mirage]

yeah but I'm choosing to emphasize this works for modules over any ring as well

[zealous berry]

i see

[quasi forum]

the base for the weak topology for example is just finite intersections of sets of the form f^{-1}(U) where U is open in the scalar field and f is in X*

it is literally just the subbasis construction

[cosmic mirage]

well also I have brainrot notation and would write Mod_k for k-vector spaces lol

[zealous berry]

anyways this was an enlightening conversation about the subbasis

thank you all

[cosmic mirage]

honestly I think it's not named well. Subbasis makes it sound like it's some ad hoc thing we put together last minute and a basis is the more important thing

but it's kind of the other way around. Subbases occur in nature very often

Its only useful to have a concept of bases because it's nice to break down constructing the minimal topology into 2 steps where you have to close it up under one operation in each step

[zealous berry]

damn really

can you frame (everything) about bases in terms of subbases?

the only thing i used a basis for so far was to show tau' is finer than tau if basis elements of tau' could locally refine basis elements of tau

[cosmic mirage]

I mean yeah basically, bases are an example of subbases

[zealous berry]

oh wait this is true

[cosmic mirage]

In the same way a topology is an example of a basis and this also of a subbasis

[zealous berry]

right

cuz topologies are closed under arbitrary union

[cosmic mirage]

yee

[zealous berry]

and finite intersection

and bases are closed under finite intersection

awesome

[cosmic mirage]

I mean I guess the analogous statement is a bit more annoying, like you have to refine a subbasis element using intersections of subbasis elements

[zealous berry]

yeah

[zealous berry]

oh this is cool too

you can consider minimality by noting how a topology drops to a basis by dropping the arbitrary union requirement, and how a basis drops to a subbasis by dropping the finite intersection requirement, leaving you with the smallest structure that can generate both

[cosmic mirage]

yeah you also have to combine this with de Morgan's laws or whatever they're called to make sure the two step construction is closed under both operations, but that's easy, it's just what I just said

[zealous berry]

reminds me of a least fixed point in type theory stuff

[cosmic mirage]

or what is it called. I want to commute an intersection past unions

whatever it is called... that thing

[zealous berry]

de morgans works

or dym A cap (B cup C) = (A cup B) cap (A cup C)

[kind marlin]

i think it's just distributivity

[zealous berry]

idk if theres a name for that, just distributivity ig

[unreal stratus]

Ye

[cosmic mirage]

cool, then yeah that thing

[kind marlin]

i wonder if there's an alternative to basis where every open set in the topology is a finite intersection of sets in the generating set, that sounds like it would be a little annoying to use

[cosmic mirage]

I mean there is some notion like that bc you can take the 1-fold intersections of just any open. But I'm worried about this being closed under unions unless your generating set is unwieldy

[gaunt linden]

Sounds like you'd need an unwieldily large generating family to get something such as the standard topology on R.

[cosmic mirage]

lmaoo nice

[kind marlin]

as in like, my thought was just take a subbasis and then perform arbitrary unions in that subbasis

but yeah this sounds generally hard to work with

[cosmic mirage]

I don't think finite intersections of arbitrary unions will be closed under finite intersections and arbitrary unions though

wait

(u_i A_i) n (u_j B_j) = u_j ((u_i A_i) n B_j) = u_{ij} (A_i n B_j)

oh okay it does work then ig nvm lol

[kind marlin]

i feel less sure now 😭 but i trust that, i will probably work it out

[cosmic mirage]

I don't trust it, lmk what you find lol

[kind marlin]

okay wait you can write an intersection of arbitrary unions as an arbitrary union of intersections which is what u showed

but you can't necessarily write an arbitrary union of finite intersections as a finite intersection of arbitrary unions

this post just says that they're not equal without proof in the question but i think its not hard to find a counterexample

https://math.stackexchange.com/questions/3511075/do-arbitrary-union-and-arbitrary-intersection-operations-commute

im happy about that at least, bases are definitely the correct way to go

[gaunt linden]

Yeah, take your bases sets to be everything of the form (-infty,a) and (a,infty). That's a subbasis for the ordinary topology on R, but a finite intersection of arbitrary unions can exclude only finitely many intervals.

[zealous berry]

You mean (a, inf)?

[gaunt linden]

Whoops yes, bothched editing. Fixed.

[zealous berry]

And it can be (b,inf) I think?

With a > b

[gaunt linden]

It needs to be all possible values of a (or at least enough different values that they're dense in R).

[zealous berry]

Yep

[empty crypt]

I have a question for a problemset on topology, how could I get help here?

[quick delta]

By asking it?

[empty crypt]

ow but can I ask it here or do I need to go to "help forum"

[quick delta]

You can ask it here

[robust drum]

This isn’t quite true though

Bases have the property that any intersection is open, (contains a basis around any point) but it need not be itself a basis

[robust drum]

I mean subbases have literally no restrictions except that they cover the space

So they’re useful in the sense that you can just force some class of subsets to be open with them

[zealous berry]

a refinement exists is what i meant by closed under finite intersection

[zealous berry]

yea

[zealous berry]

it's an intersection past intersection in this case btw

well not like anyonec ares its easy to check

[cosmic mirage]

ohhhh LOL that's the problem, thanks

cool so my proof just said that subbases work 🤡

[rancid umbra]

bases aren’t closed under finite intersection.

the intersection of two open balls in R^n is not necessarily an open ball.

oh somebody pointed this out already. whoops

[zealous berry]

Yes I meant the intersection contains a basis element

[rancid umbra]

idk if you are familiar, but there are free-forgetful adjunctions relating subbases, bases, and topologies

[zealous berry]

Does aluffi cover this

Hmm maybe when done with pointset top and some aluffi stuff I'll look into a categorification of topology

[rancid umbra]

i’m not aware of any texts that cover this. it’s not too difficult tho. similar to how subgroups are generated by subsets