point-set-topology

[prisma garnet]

oh, yea, $\Lambda$ is the union of all the faces of $\Delta^n$ except one

[red yoke]

That term would be H0(∆^0)

[gentle osprey]

DarQ

[gritty widget]

exercice: Let $f : X \to Y$ be a map between two metric spaces. Prove that the following conditions are equivalent.

(i) f is uniformly continuous.

(ii) For any any two sequences $(x_n)$ and $(y_n)$ of X, we have
$d(x_n, y_n) \to 0 \implies d(f(x_n), f(y_n) \to 0$.

solution: $(i)\implies (ii)$ suppose that f is uniformly continuous, then $\forall x,y\in A , \forall\epsilon>0,\exists \alpha>0 \mbox{ such as } \lvert d(x,y)\rvert <\alpha \implies \lvert d(f(x_n), f(y_n)\rvert <\epsilon$, if we impose $\epsilon \to 0 \mbox{ and } \alpha \to 0$ then we have (ii)

$(ii)\to(i)$ suppose that $ d(x_n, y_n) \to 0 \implies d(f(x_n), f(y_n) \to 0$, for all $x_n ,y_n\in A$ then we can say that $\forall\alpha>0 , \forall\epsilon>0$ we have $d(x_n, y_n) <\alpha \implies d(f(x_n), f(y_n) < \epsilon $ , and f is uniformly continuous

idk what i was doing

can someone help me solve this exercice

[gentle osprey]

john.

[red yoke]

Wdym if we impose ε → 0 and α → 0

[gritty widget]

like it get close to it

tbh i don't know how to solve it

[hexed steppe]

your quantifiers are wrong

review the definition of unif continuity

[gritty widget]

that is the definition that i have

i don't see where i've used it wrong

[hexed steppe]

in the single place where you used it

[tulip bluff]

In Hatcher's book on vec bundles and K-theory he keeps talking about a classifying map f : B -> G_n but does not define this anywhere in the book. What is this map?

[unreal stratus]

I would assume from context that what you write as $G_n$ is what is often denoted as $\mathrm{Gr}n$ or $BU(n)$, which is the colimit / direct limit $\mathrm{colim}{k \to \infty}$ \mathrm{Gr}_n(\mathbb C^{n+k})$ where $\mathrm{Gr}_n(\mathbb C^{n+k})$ is the Grassmannian of $n$-planes in $\mathbb R^{n+k}$ (the space of $n$-dimensional linear subspaces). There's a canonical vector bundle over $BU(n)$ which I'll denote by $\gamma^n : E(\gamma^n) \to BU(n)$. It is then a theorem that $BU(n)$ classifies complex vector bundles of rank $n$ in that for a paracompact space $X$, homotopy classes of maps $X \to BU(n)$ are in natural bijection with isomorphism classes of rank $n$ complex vector bundles.

[gentle osprey]

potato
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[unreal stratus]

Wait lol this is all in hatcher

Around theorem 1.16

[gentle osprey]

Leno

Leno

[gritty widget]

Does anyone know if vietoris rips complex can be computed fast

[last topaz]

What is a proof that if a connected subset of the punctured plane contains a loop that's non contractible in C* then it contains a generator of the fundamental group of C*?

[knotty vine]

Depends on the number of points and the dimensionality of the features. If these become large, the complex blows up at larger distances. There's tricks to remedy this though. But idk about those.

[gritty widget]

I need to make for an input of nodes (so number of nodes is not fixed)
Edges are between spatially adjacent nodes; I don't know what you mean by the distances here (like which distance)
Also what tricks? 👀

[knotty vine]

By distance I mean the distance between points for them to be spatially adjacent. If this distance becomes larger (this is what you would need to compute persistent homology for example), more and more edges/simplices get added

How many points are you expecting, like 10 or like millions?

[gritty widget]

~10^3 to 10^5

I'll be able to adjust the distance parameter

[knotty vine]

Is the intent to then compute the homology? Or do you just need the complex?

If you need the whole complex, there's no tricks (other than the standard algorithms for range searching, maybe a BSP tree or something)

[knotty vine]

Here's an idea: any path gives rise to an injective path by "cutting out" any loops. Applying this to a noncontractible loop will give you a generator. I think showing that this is indeed a generator uses the Jordan curve thm.

[red yoke]

I think you can lift to R²

And you immediately get contractible in C* iff loop and contractible in R²

[last topaz]

I think I see an elementary argument. Let $\Omega$ be the subset of the punctured plane in question. The universal covering of $C^$ is the exponential map $exp: C\rightarrow C^$. If $\gamma$ is some curve based at $z$ in $\Omega$ that's non contractible in $C^*$, let $\tilde{\gamma}$ be its lift with $\tilde{\gamma}(0)=\tilde{z}$. Then $\tilde{\gamma}$ goes from $\tilde{z}$ up to $\tilde{z}+2\pi i k$ for some integer $k$. If $k=1$, we've won because it means $\gamma$ is a generator. If not, our goal is to show that we can get from $\tilde{z}$ to $\tilde{z} + 2\pi i$ using a curve in $\exp^{-1}(\Omega)$. The key observation is that the vertical translate of $\tilde{\gamma}$ by $2\pi i $ is also in $\exp^{-1}(\Omega)$. It is a curve from $\tilde{z}+2\pi i$ to $\tilde{z}+2\pi i (k+1)$ and it is bound to intersect $\tilde{\gamma}$ so we can hop on it to get from $\tilde{z}$ to $\tilde{z}+2\pi i$.

[gentle osprey]

trout of Orms-by-Gore

[gritty widget]

im trying to find holes and their locations + chain complex to use for path finding

[gritty widget]

also i don't really understand persistent homology; some filtrations and barcodes, can't wrap my head around it

[red yoke]

Is this a #linear-algebra question

[tribal palm]

gawd i love inventing notation to simplify problems

[knotty vine]

This is nice! But how do you prove theres an intersection?

[red yoke]

If there's no intersection then you have a map I² → C* s.t. boundary has odd winding number

Twice the winding number of γ bar around z+2πi + twice the winding number around z+6πi, actually

(z-4πi? Something of the sort)

[knotty vine]

I dont understand

[red yoke]

Take the map ‾γ(t1) - ‾γ'(t2), where ‾γ' is ‾γ but shifted

[knotty vine]

Cool

[knotty vine]

Why do you need a chain complex for path finding?
Persistent homology is not that difficult if you're already familiar with normal old singular/simplicial homology and Vietoris Rips

[unreal stratus]

chain

[knotty vine]

?

You mean the chain for the hole?

[unreal stratus]

Idk i just said it

Lol

[merry geode]

Idk a lot, but it sounds like topological data analysis

[knotty vine]

Yeah, but he said he didnt understand persistent homology, so idk what the idea was

[gritty widget]

My nodes environment can move nodes around constantly, kill them/make them reappear, and I need to find possible paths from all nodes to a single node
I thought if I calculated a simplicial complex I could find out where the holes are (for coverage) and also calculate chain complex from it to find paths/update when nodes change

[bright acorn]

What are you guys favorite references on sheaf cohomology?

[plain raven]

I learned sheaf cohomology from godements book on algebraic topology and that was an absolutely beautiful book.

[umbral panther]

Chern. Complex manifolds without potential

[tribal palm]

i love writing up random expressions for no good reason

actually i am curious, does this set of the interior of the closures of singletons ever appear in theory?

[alpine nest]

in a Hausdorff space the closure of a singleton is itself, and then the interior is nonempty only if the point is isolated

so you found a fancy way to describe the set of isolated points

[tribal palm]

but when it is not hausdorff? i was specificially hoping that this collection would make the set equivalence classes of topologically indistinguishable points, without bothering to think properly about it first

it seems extremely unlikely i would guess that correctly though

[alpine nest]

if it's not Hausdorff then all bets are off

the closure of a singleton might be the entire space

[red yoke]

What happens on Spec(R)

[cedar pebble]

Suffering

[red yoke]

Is it the collection of components or sth?

[cedar pebble]

I mean Spec(R) in general has lots of non-closed points

In some cases the closure of such a point can be all of Spec(R)

[red yoke]

The closure of a point p is the collection of prime ideals containing p

Then the interior is the collection of s ∈ R s.t. s ∉ prime ideal → p ⊂ prime ideal

Wait my brain not be braining

[merry geode]

Interior of the closure of a point?

Wouldn't the open sets be too huge to be contained in the closure

[red yoke]

Not if the closure is say a component

[languid patrol]

If I take any topological space, then add one point and declare that every open in the new space is an open in the old space plus the extra point, then the point is dense in the space. This is what is going on when you pass from varieties to schemes: for every closed irreducible algebraic subset you are adding a point like this whose closure is that closed subspace

[merry geode]

Ah, irreducible component. I was only thinking about irreducible spaces and excluding points whose closure is the entire space.

[gritty widget]

HI guys im looking for a book in topology and 3d spaces

i want it to abstract the grocery lists as much as possible and sum up materiel and become from beginner to expert

qualitatevly least quantitve as possible, as least verbouse and sperflous

[red yoke]

You want it to what

[rancid umbra]

abstract the grocery lists as much as possible

[gritty widget]

thats good

ong

[tidal cedar]

Thurston prolly?

[proud pawn]

I was happily understanding the proof of Urysohn's metrization theorem and then the book gives me this image

And now I am just confused like what part of R^omega is this blobby square representing

and why is Z (which represents the image of the imbedding F) a weird squiggly line inside of it

Each vertical slice of the square gets projected onto the line I get that but then what are the horizontal slices representing, if anything???

And why the heck is it overlapping???

[radiant cedar]

When someone is talking about submanifolds representing cohomology classes are they somehow using Poincare Duality or how is this possible?

[hexed steppe]

yes its poincare duality

[radiant cedar]

Can you elaborate on this? Does every non-trivial cohomology class of a k-form correspond to a submanifold of dimension n-k?

[hexed steppe]

well yeah

modulo some assumptions on the manifold

here look at section 5 https://www.homepages.ucl.ac.uk/~ucaheps/topics/pd_notes22.pdf

[cedar pebble]

https://mathoverflow.net/questions/151430/cohomology-classes-represented-by-submanifolds

some of the discussion here might be relevant

usually you're just taking the fundamental class of the submanifold in Borel-Morre homology and then relating this to cohomlogy

but you have to be careful about issues of orientation and so on, which is discussed in the thread

if you're working with rational coefficients (or real coefficients) every cohomology class is going to be representable by some smooth submanifold; in general there are some issues with torsion

[umbral panther]

Rationally every homology class can be represented by a map from a closed manifold. But it might not be embedded. The image might have singular points that cannot be resolved

[gritty widget]

Any resources for computational algebraic topology?

[solemn oar]

Something like this?

https://monge.univ-mlv.fr/~demesma/FullLectureNotes.pdf

https://www.maths.ed.ac.uk/~v1ranick/papers/edelcomp.pdf

[knotty vine]

Also https://www2.math.upenn.edu/~ghrist/notes.html (free)

Also https://www.aatrn.net/home and their yt channel https://www.youtube.com/@aatrn1/videos

[river granite]

yeah these are all good lecture notes. wasn't aware of the yt channel

[solemn oar]

Same, this looks cool

[broken nacelle]

Whats a continuous field of tangent vectors?

Wait, nvm, it's prolly a continuous function that goes to R^n

[unreal stratus]

Depends on your manifold, but if you mean on R^n, yes

[red yoke]

Is it be Hatcher hairy ball

[feral copper]

It's a continuous section of the tangent bundle

Basically a continuous map M->TM which maps to x a tangent vector to M at x

[broken nacelle]

dunno what that is

this

[red yoke]

That implies hairy ball I think

Hairy ball theorem is the forwards direction

[opaque scroll]

I guess existence is pretty straight forward, just using the usual (a, b) |-> (b, -a) thingy

[whole matrix]

for 7.2 how does f • g^{-1} work? i thought the paths needed to have the same domain

[feral copper]

It actually is the statement of the hairy ball theorem!

A homotopy from f.g^{-1} to the constant loop can be used to build a homotopy from f to g, and vice-versa

Note that this . operation is concatenation

[gentle osprey]

Matplotlib

[whole matrix]

yea, but since f and g^{-1} have different domains would it change anything

[feral copper]

You go from p to q via f, and from q to p via g^{-1}

So it is a loop

(concatenation makes sense if f(1)=g(0) ofc)

[whole matrix]

ah i see

so if the paths were already loops the concatenation would still be a loop

[feral copper]

Yep

Going from p to p, and from p to p again

(I mean if q=p with the notations of the book)

[whole matrix]

the actual question im trying to do takes f and g in Ω(X,p) and it replaces f•g^{-1} with f^{-1} • g

[feral copper]

So they are loops in your case, right? And you want to show that the concatenation f^{-1}.g is homotopic to the constant loop iff f and g are homotopic?

Just use the above with p=q and switch f with f^{-1} and g with g^{-1}, it's a corollary

[whole matrix]

yea that makes sense, i’ll see what i can do now, thanks

if f: I -> X then is f^{-1} : X -> I or is that not the case for paths?

it must be also I -> X otherwise the concatenation wouldnt make sense, am i overthinking this?

[gentle osprey]

Matplotlib

[whole matrix]

yea i figured that, it’s slowly making more sense to me

[heady skiff]

would you just define $f$ as $\bigl((x_1, x_2, \dots, x_{n - 1}), x_n\bigl) \mapsto (x_1, x_2, \dots, x_n)$

[gentle osprey]

okeyokay

[thorny agate]

have you tried that?

does it work?

[heady skiff]

yeah i think?

[white oxide]

and?

[thorny agate]

Does the proof work?

[heady skiff]

yeah?

[thorny agate]

yea? or yea

[heady skiff]

yeah lol

i'm just tryna verify if that was the correct homeomorphism or not

[thorny agate]

if the proof works then its the correct homeo

That is the obvious map (what else would the map be) but I wanted you to actually try it in the proof before confirming

[white oxide]

I mean, if you want the "correct" homeomorphism check the universal property

[heady skiff]

yeah, i guess it was more or less a sanity check

[white oxide]

dont expect any surprises tho

[unreal stratus]

Hm smth im reading talks about a "spherical element in the homology of [a space]" - how should I interpret this?

[knotty vine]

Google says an element of the homology is spherical if its in the image of the Hurewicz map

[unreal stratus]

Oh sure that makes sense

Hm

Thank

[knotty vine]

Single thank is enough

[unreal stratus]

Lol

[alpine nest]

thank < thanks < many thanks

[sonic crane]

Guys

I tried reading first part of topology book and i was already confused and like wtf is going on

I feel like its a lot of set theory stuff and im not totally comfy with it yet idk

Basis for a topology already confusing me

[wispy veldt]

are you familiar with metric spaces and the analysis on them? or i guess my question is in what kind of space did you learn real analysis

a lot of the motivation in general topology will just be generalizing concepts you encounter in RA

[unreal stratus]

Lol is that Munkres

[sonic crane]

Ya lol

[unreal stratus]

Chap 1 randomly does lots of set theory which you can probably skip

[sonic crane]

I haven’t done any lol

[wispy veldt]

Then a lot of things wont make much sense , i highly recommend learning some real analysis with it if you are committing to topology , the definition of a basis for example naturally generalize how open intervals are a basis for the usual topology on R

[knotty vine]

What's your motivation to study topology?

There's different paths to topology, not just analysis or even geometry

[sonic crane]

idk man

to avoid depression probably

but if i dont have the background it will prob end up causing more stress and sadness

i wish my school offered another algebra course

i was enjoying that

[sonic crane]

Ok so new plan boys

Imma just not do school

[whole matrix]

if f is in Ω(X,p) then is f^{-1} concatenated with f the constant path? also is constant path concatenated with f is f right? because of the fundamental group

[feral copper]

No, it's not the constant loop, but it is homotopic to it

And no, the constant path concatenated to f is not f, it's homotopic to it

But ofc, in the fundamental group, then all those statements become true

[whole matrix]

then ig my idea of taking H as the homtopy of f and g
so G = f^{-1} • H would be the homotopy between f^{-1} • g and the constant path wouldnt work

[feral copper]

I'm sorry I'm not sure I follow; what are you trying to prove, and how are you trying to do it?

Because it seems like you're trying to concatenate a path to a homotopy, which can or can not bear meaning

[whole matrix]

show that given f and g in Ω(X,p) f and g are homotopic iff f^{-1}• g is homotopic to the constant path

my idea was (when i thought f^{-1} • f = c_p)

G(s,t)= f^{-1}(s) • H(s,t) where H is the homotopy between f and g since
when t = 0 H(s,0) = f
G(s,0) = c_p
…

[feral copper]

So do both directions separately.

• First, assume that there is a homotopy H(s,t) between f(t) and g(t). This means H(0,t)=f(t) and H(1,t)=g(t) Try to build a homotopy between the concatenate of -f and g. You may want to use H for this! Try making a drawing of course.
• Then assume that you have a homotopy between -f.g and the constant loop, and do the reverse: use this to build a homotopy between f and g.

Also, if you already have shown that the fundamental is a group with the concatenation operation, then this fact is just group theory! (really a tautology in fact)

[whole matrix]

it’s shown already :)

i’ll try to make a drawing to see if it helps me

[whole matrix]

so the issue in here is that i assumed f^{-1} • f = c_p ?

[feral copper]

Then if you have shown that the fundamental group is a group, convince yourself that you're trying to show that f=g iff f^{-1}.g=const

[feral copper]

In the fundamental group, this is true. In the loop space, not

[whole matrix]

ahhh

yea f and g are in the loop space, i’ll see what i can do now, is it alright if i ping u later for more questions?

[feral copper]

Sure

I may not be available later in the evening, but who knows, you can try

[feral copper]

In fact, it's no problem that f and g are in the loop space. f is homotopic to g iff [f]=[g] in the fundamental group, and so you're trying to prove that: $$f\sim g\iff[f]=[g]\iff[f]^{-1}[g]=1\iff[f^{-1}\cdot g]=1\iff f^{-1}\cdot g\sim\mathrm{const}$$

[gentle osprey]

Matplotlib

[feral copper]

Here ~ stands for "homotopic to"

[whole matrix]

so my idea isnt exactly wrong it’s just lacking arguments?

[feral copper]

Yup, the idea was there

[whole matrix]

ok good to know

[unreal stratus]

Hm how should one think about the classifying space construction more invariantly / homotopy theoretically? Thus far I've been able to get away with the classifying space of a discrete group, which is well defined up to weak homotopy equivalence, and you can work with a specific construction to get an actual functor, but I wondered if there is some universal property, perhaps on the level of infinity categories

[plain raven]

I'm wondering what you mean. If it's well defined up to weak homotopy equivalence it must have some universal property that characterizes it. What is the proof you know that it's unique up to weak homotopy equivalence?

And is there not a universal property implicit in that proof?

[unreal stratus]

Oh what I mean is that for discrete G, BG is just a K(G,1) and those are unique

But for topological groups and stuff it's not so simple

(afaik)

[umbral panther]

It classifies principal G bundles. That identifies it in the homotopy category. I think you can upgrade this to infinity

[plain raven]

Ok. I see.

In topology we study vector bundles and more generally fiber bundles and fibrations. A vector bundle is a map which is locally the product of a vector space with the base space and the transition maps between the local trivializations respect the coordinate systems so that linear algebra can be done in one coordinate system and transferred to another.

The idea of changing between trivializations "respecting coordinate systems" can be generalized beyond vector bundles to maps where a topological group G acts on the fibers and the transition maps between local trivializations are required to be G equivariant. We recover the theory of vector bundles when G is GL_n(V).

G bundles are fibrations btw and this is an important part of the theory.

A basic problem in the field is to classify bundles over a given space B with given fiber F and topological group G up to isomorphism. Upon studying this problem it quickly becomes obvious that F is irrelevant to the classification problem and we might as well replace it with the simplest space with a G action, namely G. Such spaces are called principal G bundles.

A morphism of bundles (p, E, B) -> (p', E', B') is defined to be a commutative square such that the top map is G equivariant.

Similarly there is a notion of homotopy between morphisms of bundles and the homotopy between top maps should be equivariant.

There is a distinguished "homotopy-terminal" bundle in this category, pG : EG -> BG such that every principal G-bundle p has a bundle map p => pG unique up to G-equivariant homotopy. Probably as bw says up to higher G equivariant homotopy.

Principal G-bundles can be pulled back along continuous maps between the base spaces so for any map f : X -> BG you can form f*(pG) over X and this has a canonical bundle map to pG.

These two last points combined means that for every principal G-bundle p over X there is a map f : X -> BG whose pullback is p, and f is unique up to higher homotopy

So Bun(X) \cong [ X; BG ]

You shouldn't focus too much on BG. The important thing is the whole bundle pG : EG -> BG.
EG also has an interesting property, it is the unique contractible space with a "nice" G action (think like, free of torsion, the quotient map is not too ugly). It can be thought of as a cofibrant replacement for the singleton in the space of G spaces with a nice action (obviously the singleton has a G action but it has a lot of torsion)

You can see Dale husemoller, Fiber bundles for more on this. Actually any textbook on fiber bundles treats this, for example steenrod.

A nice proof of the important theorem is in "partitions of unity in the theory of fibrations" by dold. The paper is a bit technical, and addresses exactly what it says in the title but among other things it gives a nice proof of this theorem stressing the connection to homological algebra and the method of acyclic models.

[unreal stratus]

Oh I mean I know about classifying spaces for bundles and somehow blanked about them for the second (sorry for wasting your time oop)

I guess I was sort of interested in the functoriality of the construction

But perhaps the point is just that any map G -> H induces a map from Bun(G) -> Bun(H) and that corresponds to a homotopy class of maps BG -> BH by yoneda

oh yeah sure and the map Bun(G) -> Bun(H) is just from (-) x_G H (induction)

[plain raven]

There are functors due to a number of people, for example, Milnor, and independently Milgram.

[unreal stratus]

I think basically I just have been thinking about operads and have been thinking of B(-) as "delooping" rather than anything with that sort of geometrical content

But they are different things in general

[unreal stratus]

Yeah there's that Milgram bar construction thing isn't there

[unreal stratus]

I think this is what I want basically

Like okay this means we go from group maps to homotopy classes of maps on classifying spaces

[plain raven]

Yes.

[unreal stratus]

and then I guess the constructions lift that to Top

[plain raven]

Milgrams bar construction is probably one of the earliest important examples of a coend

[unreal stratus]

I can't believe i blanked about BG classifying principal G-bundles for a bit and feel ashamed lol

[unreal stratus]

Oh how come

Hm

[plain raven]

I didn't say it was super important. It's modestly important and was early lol

Mac lane published a paper proving this.

[ebon galleon]

bar constructions

[unreal stratus]

Oh nice sure

I only really know about two-sided bar constructions (from May)

[merry geode]

May I ask what BG means, when G is a group? It was part of why I failed a algtop exam.

[balmy field]

Is B a ring?

If so it could be a group ring

[merry geode]

Nah, just notation

[lime sable]

classifying space of G

[cedar pebble]

yeah especially in the context of AT this should be the classifying space

[umbral panther]

B is for base, the base space for the universal G bundle

[unreal stratus]

So what is E for

Maybe it is German

[umbral panther]

[lime sable]

what if it's etale

[merry geode]

I did not skip a class, but I somehow did not learn classifying space in class

[unreal stratus]

Lol

[cedar pebble]

simply intuit

[unreal stratus]

Maybe entire

[cedar pebble]

algebraic topologists: here is a complicated infinite dimensional space that models BG
algebraic geometers: it's */G

[unreal stratus]

Don't topologists do that too

Well homotopy theory minded

[cedar pebble]

sometimes yes

[unreal stratus]

*//G

Chad me

[merry geode]

I guess prof lectured this in cohomology class which was the only class I was absent

[cedar pebble]

[*/G]

[lime sable]

BG = 1/G

[unreal stratus]

Stack

[cedar pebble]

EG=G/G

[unreal stratus]

When the thing is badly behaved so you just do a formal version

That's copium

And good mathematics ig

[merry geode]

Wh

[umbral panther]

Yes, probably a German word for total or entire

[cedar pebble]

also just a common letter to denote vector bundles

[unreal stratus]

Lol

[dry jolt]

It's because the letter E is contractible

[unreal stratus]

E is overused

[cedar pebble]

8 = S^1 v S^1

[umbral panther]

It might be because E is near in the alphabet to F, for fiber

[unreal stratus]

EHP, E_n algebra, Lubin Tate theory E_n, E(n), ... lol

Total space of a bundle generically

Generic name for a cohomology theory

[merry geode]

So what defines the bundle map EG -> BG ?

[cedar pebble]

it's the universal G-bundle

[unreal stratus]

Basically you take a contractible space EG on which G acts freely

[merry geode]

I mean the choice of EG

[unreal stratus]

Then BG = EG/G

[merry geode]

Ah, I see

[cedar pebble]

EGG

[unreal stratus]

For example for G = Z/2 you can take S^infty with the antipodal action

[merry geode]

Sounds like K(G, 1) is relevant

[cedar pebble]

it literally is a K(G,1) yes

[unreal stratus]

For discrete G you get a K(G,1)

In general it isn't so easy

e.g. BS^1 = CP^infty

[merry geode]

IIRC the exam was about discrete group

[unreal stratus]

Which is a K(Z,2)

[ebon galleon]

something something enriched bar construction

[unreal stratus]

lol

How to tell someone read riehl

You and moldi have these vibes

[merry geode]

So the prof stumped me by using terminology I don't know

[unreal stratus]

I need to read it

Sleepy times now. Goodnight.

[ebon galleon]

I'm really liking it but I also really need to sit down and work through some of the details more

gn 🥔

[unreal stratus]

Me with all maths

[plain raven]

My message above is an explanation of what BG is

[plain raven]

Riehl's book is good. The stuff in it that refers to Garner's factorization systems is outdated though. Garner published a really good paper a few years later that makes the older paper obsolete

Ask me how I know this.
|| I accidentally reproved several of Garner's theorems from said really good paper because i had no idea it existed ||

[cedar pebble]

let me guess the explanation is an entire page long
yup

[merry geode]

Ahh.. sorry

[balmy field]

https://tenor.com/view/gigachad-chad-gif-20773266

Yo urn

[ebon galleon]

Noted, thanks

Not sure if I was gonna read that section anyways but nice to know of good papers

Also been reading some of Shulman's paper on homotopy limits, is quite nice

[plain raven]

I have read some of this paper but I did not finish it. I got enough through it to get some interesting ideas out of it

A certain ethereal homotopy type theory student recommended it to me

she lives up in the clouds and comes down to answer mortal questions once in a while

fascinating

[ebon galleon]

Holy shit

That is my aspiration

[foggy portal]

Are modern algebra and topology necessary prereqs to take a first course in algebraic topology?

The professor claims that these prerequisites won’t be needed but I’m a bit wary.

[fading vale]

you need to know the basics of what would be covered in either course and be comfortable with a lot of the content in them

it would be wise to take them first

[foggy portal]

Would a course like that use more alegbra topics, more topology topics or a fair mix of both?

In other words, if I were to only take one of those courses which would be more worthwhile

[ebon galleon]

I mean that would depend on what you already know

[foggy portal]

I haven’t taken modern algebra or a general topology course, so not much. I’ve done measure theory but I doubt that overlaps here

Sounds like it wouldn’t make sense to skip steps here. I’ll do algebra and topology first

[quiet thorn]

you should probably at least take a group theory and general topology class

oh im 2 hours late

[cerulean oriole]

Consider the category of topological spaces with morphisms being formal signed sums of continuous functions (i.e., apply the free abelian group functor (which preserves monoidal products) to all the homsets of the usual category of topological groups to get a category enriched over abelian groups).

In this category, we have a cochain complex $(C^n, d^n : C^n \to C^{n+1})$, where $C^n = \Delta^n = [e_0, \dots, e_n]$ is the $n$-dimensional standard simplex, and $d^n = \sum_{i = 0}^{n+1} (-1)^i [e_0, \dots, \hat{e_i}, \dots, e_{n+1}] \colon \Delta^n \to \Delta^{n+1}$, where I'm going to hope you can guess what $[e_0, \dots, \hat{e_i}, \dots, e_{n+1}$ means without my explaining it.
(For $n < 0$ take $C^n = 0$ and $d^n = 0$.)

Then for any topological space $X$, the singular chain complex of $X$ is the image of this cochain complex under the contravariant functor $\operatorname{Hom}(\mathord{-}, X)$ (valued in abelian groups).

This suggests that this cochain complex is in some sense the "universal" complex behind singular homology. (Indeed, some of the basic results like the prism construction showing that homotopies of maps induce homotopies of chain complex homomorphisms seem to in some sense carry out an argument for this universal cochain complex and then use the Yoneda embedding.)

\renewcommand*{\labelenumi}{(\roman{enumi})}
\begin{enumerate}
\item Is this perspective useful?
\item Are there other cochain complexes of topological spaces which the $\operatorname{Hom}$ functor is applied to obtain a well-known homology theory?
\end{enumerate}

[gentle osprey]

Raghuram

[unreal stratus]

To me this seems closer in spirit to talking about simplicial sets

[prisma garnet]

I don't understand how he got to this conclusion

[umbral panther]

Look up the Dold-Thom and Dold-Kan theorems

[tribal palm]

planning to ace this topology exam mwahaha (i haven't gone through even half the material and the exam is in 10 days)

[ebon galleon]

You will do great

[alpine nest]

You're much more prepared than most students then!

[tribal palm]

swag

i worked through everything in the first couple of sections of Viro's general topology textbook over the last two weeks, but though it was going well it was also going way too slowly

i need to learn to prioritize or omit certain things

so now i'm just going to work through all the prof's recommended exercises in munkres instead, which is way way less work than viro and should be perfectly doable if i can just work somewhat consistently

[white oxide]

You got this, lets do a quick warm up

Write down the entire proof of tietze-urysohn from memory

(When was the last time someone had to use metrization theorems actually?)

[wispy veldt]

What kind of metrization theorems?

[white oxide]

Like urysohn and the other one

[red yoke]

The munkres book metrization theorems

[white oxide]

A space with property P (purely toplogical) is metrizable

[cerulean oriole]

IDK about metrization theorems themselves, but I have seen the fact that a countable product of metrics is metrizable, mainly in two places:

• If a topological space has an exhaustion by compact sets (https://en.wikipedia.org/wiki/Exhaustion_by_compact_sets), then uniform convergence on compact sets is given by a metric (uniform convergence on each compact set K is given by the sup_{x in K} |f(x) - g(x)| metric).
  I vaguely remember this being relevant in complex analysis (uniform convergence on compact sets preserves holomorphicity, so it's relevant, and metrisability relates compactness with sequences having convergent subsequences).
• Two definitions of Fréchet space in functional analysis are a complete space with topology defined by a countable family of (separating) seminorms, and a complete space wrt a translation-invariant metric.
  You don't need to know the details if you don't want to, but a seminorm gives a (pseudo)metric and to go from the countably many seminorms to a single metric, one uses the same idea.

Technically neither of these use the fact that a countable product of metric spaces is a metric, but the (apparently?) stronger property that the topology generated by countably many (pseudo)metrics is (pseudo)metrizable.

Also don't quote me on the first one, though I remember it being mentioned around the time we saw the Riemann mapping theorem, I don't remember how or if it was actually used.

[white oxide]

Yeah I can see those ones, mostly because the metric you obtain derives from other already understood metrics

[wispy veldt]

Well i have to check other metrization theorems but urysohn is highly useful in certain areas , existance of cut-off functions in R^n is widely applicable in distribution theory for example. A consequence of urysohn which is partition of unity is also a important tool that shows up in Reisz-Markov representation theorem and many other areas.

[white oxide]

I see

[cerulean oriole]

The one way I have seen metrizability used is equating compactness with sequential compactness (in functional analysis). So that's the only possible application I can offer.

[white oxide]

I am not analyticpilled enough to appreciate it in its full extend 😔

[cerulean oriole]

Oh, that's another example (but again only the countable product is metrizable one): the weak* topology on the continuous dual (or is it just its unit ball?) of a separable normed space is metrizable.

[wispy veldt]

this is what i was talking about , extremely useful.

Analysis aside its also useful in differential geometry ( most likely Diff topology too but i cant speak on that)

[white oxide]

Its so useful ive seen even the name in AG

Is the partition of unity thing used to prove the metrization theorem or was it a corollary, I dont remember

[cerulean oriole]

Urysohn's metrization theorem only uses Tietze's extension theorem AFAIR.

[white oxide]

Okay its used in some way in smirnov's theorem

[hidden crag]

isn't that the vodka

[white oxide]

Although I still think

man screw you :(

Although I still think I prefer the drinks over his theorems

[tribal palm]

i don’t even remember the statement rn, lol

[alpine nest]

I can't even spell Urysohn from memory

[tribal palm]

shouldn't there be an easy proof just using the urysohn lemma?

oh that is the proof given in wikipedia

ok not so easy then

[distant lichen]

Partitions of unity are the standard tool to deal with any sort of sum or integration over a space

[white oxide]

It is a lemma in the category of topological spaces that has the universal property of me deciding ill try to go through the proof tomorrow instead

[tribal palm]

lovely

[proud pawn]

Guys I'm a little confused by the proofs of "every regular space with a countable/countably locally finite basis is normal". Why is it necessarily the case that $\bigcup_{i=1}^{n}\bar{V}_{i}$ is closed?

[gentle osprey]

Minimarshoo

[proud pawn]

Couldn't it in some situations be an open F delta set?

Or wait... I just wrote the answer to my question in latex already.

[tribal palm]

is this notation used at all or did it come to me in a dream

[ \prod X_J := \prod_{\alpha \in J} X_\alpha ]

[gentle osprey]

Jens

[feral copper]

That's niche but I've seen it before

[tribal palm]

thanks, then i probably have too, though i've no idea where

[unreal stratus]

don't think i've ever seen that

more common just to supress the set and just say \Prod X_alpha

if you need to use a shorthand

[tribal palm]

oh ye that seems much more sensible

[umbral panther]

That notation is awful. People who want to be really short write X_J

[unreal stratus]

Yikes

lim X_\bull

[umbral panther]

A_J is very common with adeles, but there is inconsistency in whether J are the included coordinates or the excluded ones

[tribal palm]

what do you mean by included or excluded

oh is it perhaps something specific to adeles?

[tribal palm]

i love using slightly odd notation for simplifying expressions

[gritty widget]

I really really need some lemma to prove a result i have issues with, but I am not sure if it holds and how exactly to show it. I am trying to prove the following lemma:
Suppose we have some topological space $X$ (which is perfectly normal, but we might not need all of that to prove it). Then, suppose we have a closed set $B$ and a closed set $A$ in $X$, such that $B \subseteq A \subseteq X$. Further, suppose we have another set $C$, such that $\overline{C} \cap A \subseteq B$. Can we find an open neighborhood $W$ of $C$, for which $\overline{W} \cap A \subseteq B$?

[gentle osprey]

matcool473

[safe torrent]

One step closer to crankery /s

[tribal palm]

[gritty widget]

If it isnt true i would also appreciate just a counterexample

[wispy veldt]

regular is enough , take interior of C and find W with closure subset in interior C and conclude.

[gritty widget]

But then W does not include the entire C if C is not open?

I dont think i understand what exactly you mean

[wispy veldt]

oh my bad i read W contained in C

[white oxide]

W is meant to be open in X?

The hypotheses are too weak in that case, arent they? Let A=X= real numbers, and B=C=[0,1], then W would be forced to be [0,1], hence not open

[gritty widget]

Hm yes in my case we can assume that A has empty intersection with C

[umbral panther]

Yes, specific to the adeles. you have a fixed product indexed by I and then you take products over subsets J of I

[tribal palm]

right

[fading vale]

Let X be R^3, A the plane z = 0 B the line y = 0 in A, C the plane y = 0 with B removed. C is disjoint from A, it's closure intersected with A is just B, but any open neighborhood of C intersects A \ B

[merry geode]

Huh, adeles?

[white oxide]

Look away absta, they are trying to corrupt you with their number theory

[merry geode]

Ah, restricted product over local fields

[white oxide]

Nooo

[manic temple]

quick question: what's this | notation that hatcher's using?

[white oxide]

Restriction of domain

[manic temple]

thx

[sand dust]

Does closure and inside of set cancels?

[tribal palm]

consider [0,1)

[unreal stratus]

Or {0}

[sand dust]

wairr thats and open or a closed set

[unreal stratus]

Neither

(In R)

By inside I assume you mean interior

[sand dust]

Yess

[sand dust]

true thank u, so it cancels only if the starter set is a closed or open set

[tribal palm]

first elaborate on exactly what you mean by “cancel”

[sand dust]

closure of interior of A = A
Or the oposite

[tribal palm]

consider (0,1) what is the closure of it’s interior?

[sand dust]

[0,1] 💀

okay i messed up it all thank u guys for examples

[tribal palm]

you’ll probably want to look into the relationship between closedness and closure, openness and interior

also the boundary of a set, which is defined to be the complement of the closure with the interior

[sand dust]

Ive exam in 2 days ill check out thanks for advice :v 🫶

[knotty vine]

Here's a funny problem: In a general topological space, how many distinct sets can be created by repeatedly applying only the closure and interior operations on some given subset?

[white oxide]

no idea, more than 3... 14?

[ebon galleon]

Kuratowski

[knotty vine]

That's for closure-complement cheater!

[white oxide]

I have no idea what you are talking about

whistles innocently

[ebon galleon]

This should be phrased as "what's the maximum number of distinct sets ....", as if your set is clopen this is trivial!

[red yoke]

0, (1, 2)(2, 3), Q∩(3, 4)
0, [1, 4]
(1, 4)
[1, 4]
(1, 2)(2, 3)
[1, 3]
(1, 3)

Maximally by kuratowski

(Boundary is monovariant)

[tender trail]

Anyone here experience in weak topology?

[blazing harbor]

I'm interested in getting a better understanding of how to use math to map a planar 3d model which does not contain any holes to best fit a square with minimal distortion. I'm hoping someone might have expertise in this area and be able to help? Thank you! Also let me know if this is the correct place to be asking a question about 3d space manipulation, or if that is a different course?

[tender trail]

If you have a functional f: X to E with E being a Banach space equipped with the weak topology sigma(E, E star) and X is a topological space.
To show f is continuous at x in X, do you need to show for all epsilons > 0, for all natural number k, and for all {g_1, …, g_k} in E*
There exists delta > 0 such that for all y in X if | x - y |_X< delta then for all i in {1, …, k}
|g_i(f(x)-f(y))| < epsilon?

[naive trench]

Cant use epsilon delta

That is for real or complex

Need to use the definition of continuity at a point from topology

[naive trench]

Btw whats f_i?

[tender trail]

f_1, …, f_k is a finite set of functions in E*

So for all i in {1,..,k}, f_i is in E*

[naive trench]

Ohhh

In any case I think you need a norm in X or at least be a Banach space. Other wise you need to use the definition of continuity from topology not epsilon delta

[tender trail]

Ok thanks

[hexed steppe]

what

ignore this person @tender trail

[tender trail]

X in my problem is a topological space so it comes with a norm
When i say for all y in X such that | x -y |_X i mean in using the X norm

[tender trail]

So what i said before is correct?

[hexed steppe]

wait no

i misread your question

do you mean X is a topological vector space

[tender trail]

Yes

[hexed steppe]

ok

that means something very different

[tender trail]

Wait no i mean X is just topological space

[hexed steppe]

what

ok nvm alex is right

[fading vale]

Journeys today in the chat

[hexed steppe]

you didnt even reference f here

[naive trench]

Then its a normed space not a topological one

[naive trench]

What? Im confused now

[tender trail]

I made a typo

[hexed steppe]

now it is somewhat more plausible but

again if X is an arbitrary topological space

“|x-y|” is meaningless

[tender trail]

If X is a topological space shouldn’t it have a metric?

[hexed steppe]

no

[tender trail]

Okay

I mixed up my topological spaces with normed spaces then

Sorry for that

[balmy field]

Those that induce a metric are called metrizable

[hexed steppe]

you should in fact ignore me and just read what alex said initially

namely that you need to use the definition of continuity

for arbitrary topological spaces

[naive trench]

From a metric space you can make a topological one. From topologic to metric no

[tender trail]

Ok that clears things up

Sorry for the confusion everyone, but thanks for the help.

[naive trench]

Its ok dw

[sullen nimbus]

i am having trouble with 12.3x

how are they supposed to be related

Fr A here means boundary of A

[lime sable]

12.2x => 11.26
notice that int A, int X-A, ∂A partition X. if F ∩ ∂A is empty, then we can define F -> {0, 1} by pasting constant maps defined on F ∩ (int A) and F ∩ (int X-A) (both open in F) that are 0 and 1 respectively. but the image of a connected set is connected, so WLOG F does not meet int A, and since it doesn't meet ∂A, it doesn't meet A.
11.26 => 12.2x
let f -> {n_i}_i be locally constant on the connected subset F of X, where the n_i are distinct points of a discrete space. then the fibers f^-1(n_i) are disjoint open (clopen, even) subsets whose union is F. but since F is connected, there can only be one value of i such that f^-1(n_i) is nonempty. therefore the image of f is actually this particular n_i, meaning f is constant. (you can abuse 11.26 instead of using this fact about connected sets by setting A = f^-1(n_i) and concluding that it's empty or F because it doesn't have a boundary)

[sullen nimbus]

right! thank you

[tribal palm]

so swag Viro

[merry geode]

Does anyone know about Jones polynomial? I wanna gain insight about its cursed nature

It comes from U_q (SL2C), how does one even deal with it?

[distant lichen]

https://arxiv.org/abs/1103.5628

This is a good introduction to the topic

There's also a book by John Baez called Knots, Gauge Fields, and Gravity that covers the development of some of these ideas

In the context of Chern-Simons observables reproducing the Jones polynomial

[tribal palm]

i just realized i have, without thinking, gone from describing maps between topological spaces by how they act on points, to describing them by how they act on sets

points are such a bother

this is a very odd experience though

i catch myself writing down expressions that i have not even defined, yet they make perfect sense to me

[cedar pebble]

now you are ready for pointless topological spaces

[white oxide]

You mean half of the examples in a point set course

[tribal palm]

[cedar pebble]

spaces with no points but which are distinct from the empty space

[tribal palm]

unfortunately i am not sure that would help me pass my top exam

[white oxide]

Scare away the professor with category theory and algebraic structures?

[tribal palm]

my prof is a wiz in cat theory

he told me he met maclane himself

[white oxide]

Don't believe you, send 200 cat pics into my DMs

[tribal palm]

so i am only two degrees of social separation away from maclane mwahahaha

[tribal palm]

i forgot to mention that i am afraid of cats

[white oxide]

You have a maclane lane of 3

[tribal palm]

[white oxide]

Highway lanes usually aren't indexed starting at zero

[tribal palm]

i wouldn’t know, i’m a proud pedestrian

[white oxide]

How is the prep going anyways

[tribal palm]

horribly

have been sitting at my desk for four days without being able to concentrate

[tribal palm]

instead of moving onto topics i have to cover i keep inventing arcane notations

[white oxide]

whats are the circles for?

[tribal palm]

the set of unions/intersections in a collection of sets

as opposed to the union/intersection over a collection of sets

[woven halo]

Oh so it’s just the definition of a topology generated by a subbasis ?

Product topology ?

[white oxide]

Yeah

[tribal palm]

yes

i cheese the indices a bit though

[white oxide]

personally id just put the < and > on the outside

[tribal palm]

i use single < >s in the manner of bourbaki to denote direct (or inverse) images of sets, they are double since it is a direct image of direct images

[white oxide]

direct image of what?

[tribal palm]

the topology is on $\prod_{\alpha\in J} X_\alpha$, where each $X_\alpha$ is generated by a basis (\mathscr B_\alpha)

[gentle osprey]

Jens

[tribal palm]

and $\pi_\beta\colon \prod_{\alpha\in J} X_\alpha\to X_\beta$ is given by $x\mapsto x_\beta$

[gentle osprey]

Jens

[tribal palm]

so the repeated J subscript above is just a cheesy shorthand for...

[red yoke]

When Einstein notation

Rip I still don't get the angle brackets

[tribal palm]

so even though pi_beta is really a function from prod_alpha X_alpha to X_beta, i write as if it were a function from P(P(prod_alpha X_alpha)) to P(P(X_beta)

[red yoke]

[tribal palm]

...which is why i emphasize with the angle brackets what we are doing

[red yoke]

How about (f*).

To differentiate (f.)* and (f*).

[tribal palm]

what is (f.)* and (f*).

[red yoke]

Where f: A → B, f.: P(A) → P(B), f*: P(B) → P(A)

[tribal palm]

oh

[white oxide]

P(A)?

[tribal palm]

i don't mind the ^-1 notation

[red yoke]

Power set

[white oxide]

yikes

[red yoke]

6 function formalism

[white oxide]

The what?

[distant lichen]

A 6 functor formalism

It phrases cohomology as assigning derived categories of sheaves to spaces

[white oxide]

I mean the power set doesnt really have anything to do with that

[distant lichen]

Ahh, yes.

[white oxide]

Okay that was probably their joke in hindsight

[unreal stratus]

Lmao 6 functors

[knotty vine]

I can never remember where all the symbols go f^*, f_*, f^!, f_!, f_????

Then again I cant even remember if the index of a cochain complex is subscript or superscript

[woven halo]

Superscript

Usually contravariant stuff is in superscript, covariant in subscript and I have no idea why

[knotty vine]

oh ig it makes some amount of sense for the yoneda embedding y : C -> Set^(C^op) cause then y^c_d are morphisms from c to d

thats not too bad actually

Wait, is the standard n-simplex written as Delta^n or Delta_n ?

Delta^n would make more sense in that case I think

[white oxide]

Have you worked a bit with the pushforward and inverse image functors in AG? Once you rigidify a few in your memory things should to fall into place

[knotty vine]

tbh I don't know squat about ag. I only know the notation from Kan extensions. But it works in a topos too right? I should learn more about those...

[nimble portal]

wtf is that last thing supposed to be

I can't visualize it

I understand up until then though

Ohhh I see

You need to "glue" so the arrows have the same orientation

Hence the twist in the middle

How do you define orientation? Is the Mobius strip topologically just a square with two edges of opposite orientation glued together?

[gritty widget]

How does one "visualize" a motivic spectrum?

[cedar pebble]

same way you visualize spectra

just more complicated

idk what sort of answer you want lmao

[gritty widget]

I dunno lol

[odd gulch]

Is a set compact iff it has a finite amount of non limit points

That’s a defn this book is giving me I’m unfamiliar with

[red yoke]

Perhaps there be context?

[odd gulch]

“Topology of plane sets” by Newman.
The definition reads “13. The set E is compact if every infinite set of points in E has at least one limit point in E”
This reads the same to me as containing a finite amount of non limit points, for if you contained an infinite amount you’d be able to construct an infinite set of non limit points.

It’s a chapter on the topology of sets of points.
I really don’t know much about topology beyond some basic real number stuff like Heine Borel thm and whatnot. The definition I’ve read before is “every open cover of E has a finite subcover also covering E”, I was wondering if these two definitions were equivalent.

Also lmk if wrong channel I know this is super basic

[red yoke]

On a metric space (eg subset of a plane), compactness is the same as limit point compactness: E is compact iff every infinite set of points in E has a limit point

I don't think this is equivalent to having finitely many "non-limit points"

Compactness implies there are finitely many isolated points (but not the other way around), if that's what you mean

Or are you referring to the limit points of E in the plane?

Then all limit points of E are in E itself since compact sets in the plane are closed

[odd gulch]

Waiittt

[odd gulch]

I misunderstood it as every infinite subset of E contains a point that is a limit point of E; does it actually mean that from every infinite subset of E you can construct a sequence that converges to a point in E?

As in the subset itself contains the sequence of the limit point, not necessarily the limit point itself like I previously understood

[red yoke]

Yes

[odd gulch]

Ah cool

Checks out. Thanks!

[radiant cedar]

What is the intuition behind the cohomology ring of S^n being Z[x]/(x^2)?

[red yoke]

Well the cohomology groups are H0 = Z, Hn = Z, and 0 elsewhere

The only choice is Z[x]/(x^2)

[tribal palm]

waw, a good space

[white oxide]

give it a little pat

[tribal palm]

ok so this is something i desperately need cleared up...

is $U\sqcup V$ just a shorthand for $U\cup V$ where it is understood that $U\cap V = \emptyset$ ?

[gentle osprey]

Jens

[naive trench]

Yeah

[tribal palm]

then why have i also seen people define something along the lines $U\sqcup V := U \times {0} \cup V \times {1}$

[gentle osprey]

Jens

[white oxide]

these are slightly different notations, you should try to use only one for each

$\sqcup$ for the former situation and $\coprod$ for the latter

[gentle osprey]

Kerr

[tribal palm]

i see

[knotty vine]

Eh it's not really a problematic ambiguity imo. Strictly speaking the first meaning is somewhat bad but whatever

[unreal stratus]

Hm often they mean the same thing though lol

Like to me the first notation just means "disjoitn union of sets" and typically that is modelled as Jens describes for \coprod

[white oxide]

Yeah no commonly agreed to do it

I just like \coprod more and the square union symbol ive seen most of the time when its actually a regular union that just happens to be of disjoint sets

[tribal palm]

well in the second def U x 0 and V x 1 are always disjoint even if U and V are not

[unreal stratus]

Okay basically I disagree with the first usage you mentioned lol

[tribal palm]

aha

[unreal stratus]

But yeah often like it doesn't matter as you will get homeomorphic things either way

Really the particular construction of the disjoint union doesn't really matter

[tribal palm]

my prof introduced them in the context of connectedness

defining a space to be disconnected if it can be written as a nontrivial disjoint union of open sets

but it is not clear to me what is a nontrivial disjoint union with the second def given above

[unreal stratus]

Yeah okay that is potentially disleading lol, like in the first one it should be a disjoint union in the sense of sets

as in a union of open sets which are disjoint lol

[tribal palm]

[unreal stratus]

In the second it'd mean that the space is homeomorphic to a disjoint union of two things

But yeah

Really it should be "it is a non-trivial union of disjoint open sets"

Well, unambiguously

Note there is a real difference here because like

Connected components of Q are just singletons

But Q is not homeomorphic to a disjoint union of tons of singletons

Lol

[red yoke]

What if you just X ≈ A amalg B

[unreal stratus]

amalg lol what is that

[red yoke]

$\amalg$

[gentle osprey]

Arki

[red yoke]

Cuz my keyboard sucks

[unreal stratus]

Oh I always write \coprod lol

[red yoke]

$A \coprod B \amalg C$

[gentle osprey]

Arki

[unreal stratus]

okay true

Nice

TIL

[tribal palm]

phew ok thanks, i was worried there was some sort of equivalence i was missing here

[cerulean oriole]

In this case, not really, because of the openness assumption.

If X is a topological disjoint union of topological spaces, then their subspace topologies when viewed as subsets of X are the original topologies and it is the disjoint union of those subsets, which are also open.

Conversely, if X is a disjoint union of subsets which are open, then it can be shown that it is the topological disjoint union of those subsets with the subspace topology as well.

[unreal stratus]

Yeah agreed

[feral copper]

$$A\sqcup B$$

[gentle osprey]

Matplotlib

[tribal palm]

for showing R \ Q with the subspace top is disconnected, it's literarily as easy as just saying ((-infty, 0) n R\Q) u ((0, infty) n R\Q) is a sep of nontrivial, disjoint, open sets with union R \ Q ?

[white oxide]

yuh

[tribal palm]

lovely

i have decided i find munkres' exercises lame and will rather just do my own selection of problems from viro in preparing for my exam

[white oxide]

munkres on top of his piles of book sale money hearing this:

[tribal palm]

i mostly enjoy his exposition though

[ebon galleon]

Munkres hater

[tribal palm]

but what the heck does viro mean by this exercise "characterize the disconnected subsets without mentioning the relative topology"

how can one talk about subsets being disconnected or not without considering them as a subspace

[radiant cedar]

Why is this the only choice? If the ring is the direct sum of the cohomology groups why isn't it just Z + Z from H^0 and H^n?

[unreal stratus]

That is a description as a group

[last topaz]

Let's say X is a topological space and A is a subset of X. Give a necessary and sufficient condition for A to be disconnected (in the subspace topology) in terms of only open and closed subsets of X.

[gritty widget]

If I have a closed interval I = [-1,1] and J = [-1,1], then how is I × J a compact cylinder (visually), with I × {1} and I × {-1} being upper and lower bases respectively?

What I know of is that the Cartesian product of two closed intervals yields a rectangle on the xy-plane, in this case a square with corners (1,1), (1,-1), (-1,1), (-1,-1)

[lime sable]

is there more context? in topology, a cylinder of a space is the product of the space with the interval

[gritty widget]

I was reading up a proof on Picardo Lindelof Theorem, where (t_0,y_0) is the initial condition

And f is a function of t and y (y is dependent on t)
[Differential equation is y' = f(t,y(t))]

[gritty widget]

So like in this case too we get a rectangle with the corner points (t_0 - a, y_0-b), (t_0 - a, y_0+b), (t_0 + a, y_0-b), (t_0 + a, y_0+b)?

[radiant cedar]

wdym? Isn't Z + Z a ring too?

[red yoke]

It's a graded ring

The generator α of H^n times itself is an element of H^2n, so it is trivial

A generator of H^0 is the identity

And that gives Z[α]/α²

[hidden crag]

remember that the ring structure isn't arbitrary but comes from the cup product

[unreal stratus]

Well the nornal ring structure on Z + Z is not compatible with the grading we have here

[tribal palm]

oh ok, thank you!

[tribal palm]

waw is the proof for arbitrary products of hausdorff spaces is also haussdorf super easy or did i do something wrong

[gleaming warren]

looks easy since you have a pretty concrete description of a basis of the product topology

[graceful abyss]

In the closed singleton caracterisation, {(xi)} = product of the {xi} is closed for the product topology as a product of closed sets

[gleaming warren]

wdym?

[graceful abyss]

Which part?

[gleaming warren]

what do you mean by this? what is the context

do you want to prove this or do you suggest this?

[graceful abyss]

Wait I'm an idiot, closed singletons is an equivalent caracterisation of T1, not T2

[gleaming warren]

closed singleton does not imply Hausdorff

[graceful abyss]

Yeah, it's T1 and not Hausdorff (T2)

[gleaming warren]

okay

[unreal stratus]

Hausdorff is equivalent to the diagonal being closed and the product of closed subsets is closed

That's one nice way ig

[tribal palm]

i’m struggling a bit with this one: $X$ is Hausdorff iff for each $x$ we have
$${x} = \bigcap_{U\ni x} \text{Cl}\ U.$$

[gentle osprey]

Jens

[tribal palm]

i have the forward direction

but it’s so weird to me we take the intersection of all sets containing x

[hexed steppe]

why is it weird

[gleaming warren]

For the converse suppose that $y\in \bigcap CL U$ with $x\neq y$ and derive a contradiction

[gentle osprey]

Max

[hexed steppe]

you dont need to do that

the hypothesis implies cl({x}) = {x} for all x

[gleaming warren]

well its one way of doing it

[tribal palm]

just not used to it

[hexed steppe]

wait nvm you do need to do that maybe

[gleaming warren]

this is somehow $\subset$ am i right?

[gentle osprey]

Max

[hexed steppe]

what does that mean

[gleaming warren]

it shows that the LHS is contained in the RHS

[hexed steppe]

i dont follow

what shows that

[gleaming warren]

no i think your argument is right

[hexed steppe]

and LHS/RHS of what

no i dont think my argument is right lol

like the assumption does imply singletons are closed

but i dont think that does anything

[gleaming warren]

since cl{x}={x} you intersect with {x} and the intersection is contained in every set that you intersect with

[hexed steppe]

huh

sorry i still cant tell what youre saying

[gleaming warren]

Well U ={x} is a choice of a set with x in U. hence clU={x} appears in the intersection, hence the intersection is contained in {x}

[hexed steppe]

what statement are you describing a proof for

[gleaming warren]

For $\bigcap_{x\in U}cl(U)\subset {x}$

[gentle osprey]

Max

[hexed steppe]

huh

the setup is

assume they are equal

why is it necessary to prove that containment

[gleaming warren]

Oh my fault i read let X be hausdorff show that ...

[hexed steppe]

oh lol

yeah then i agree with you

but anyway you can try to use the hypothesis to produce a separation of points

[gleaming warren]

Suppose X is not Hausdorff then there are x y that cannot be separated by open sets hence for every closed C set either x and y are in C or both not

[hexed steppe]

yeah

[gleaming warren]

this should imply that $\bigcap_{x\in U}cl(U)\supset{x,y}$

[tribal palm]

ok so for p ≠ q, by hypothesis {q} = the intersection above, and since p is not in {q} there is some S with p not in the closure of S, and we can let U = X \ Cl S, which is open and contains p… now if only S were open

[gentle osprey]

Max

[hexed steppe]

is every closed set the closure of an open set?

oh ok you edited

[hexed steppe]

what is S

isnt the intersection in the problem statement taken over open sets U containing x

[gleaming warren]

over any sets

[unreal stratus]

Isn't this false or am I dumb like

If you are T1 doesn't this hold

[graceful abyss]

Really?

[tribal palm]

it might be that Viro is taking it as implicit that U is open

[gleaming warren]

the exercise is that this is equivalent to being hausdorff

[graceful abyss]

In my mind it should be closed neighbourhoods

[unreal stratus]

Yeah sure the notation implies that ig

[graceful abyss]

So U open

[hexed steppe]

i mean

[unreal stratus]

I know lol

[gleaming warren]

idk i just read how he shared it

[hexed steppe]

obviously the intersection over all sets

[unreal stratus]

I'm saying it seems wrong as stated

[hexed steppe]

is thr singleton

[tribal palm]

that U should be open?

[hexed steppe]

yes

[unreal stratus]

Yeah

Otherwise this is equivalent to T1 if you use all sets containing the singleton not just open ones

[hexed steppe]

the hypothesis trivially implies singletons are closed

[graceful abyss]

If you take all sets, then there's in particular just {x}

[unreal stratus]

Yes lol

[hexed steppe]

singletons being closed is equiv to T1

[unreal stratus]

If t1

[tribal palm]

this is how Viro states it (at the bottom of the page)

[hexed steppe]

that is like also trivial

[unreal stratus]

Yeah should be open U

[graceful abyss]

Yeah he's lazy but U practically always means open

[tribal palm]

i am also rather confused by the similarity with 14.11 here two pages after the first

[tribal palm]

phew ok then i have it, thanks for clearing this up

[gleaming warren]

this is frequently used in many areas but i dont like the concept that a specific letter always means open or e.g. kappa always means cardinal

[unreal stratus]

Agreed

[hexed steppe]

what do you mean

one is closure

one isnt

[tribal palm]

oh

[hexed steppe]

they are different statements

[unreal stratus]

UwU

[tribal palm]

*insert jens cheems flush sparkles*

[hexed steppe]

anyway most spaces in real life are either polish or not even T1

[unreal stratus]

Lol

[tribal palm]

hwat is a polish space

[unreal stratus]

Is every cw complex polish

I guess no due to a potential lack of finiteness

[hexed steppe]

separable and completely metrizable

[unreal stratus]

Like can just do a massive discrete thing lol

[hexed steppe]

algebraic topology isnt real life

[unreal stratus]

Lol

Depending on what you mean by real life

[hexed steppe]

ok let me revise my claim: everything is metrizable or spec(R)

[unreal stratus]

Or a scheme

But yes sure locally

[gleaming warren]

Certainly in algebra but in analysis also other things arise

For instance the maximal ideal space of a Banach algebra

[hexed steppe]

oh is that not metrizable

[gleaming warren]

i would guess no in general

[gleaming warren]

Oh yeah for instance the maximal ideal space of l infinity (little l) is the Stone-Cech compactification of the natural numbers and therefore not metrizable

[graceful abyss]

The space of test functions is also not metrisable

[hexed steppe]

oh ok

ok thats an important one too

fine

everything is hausdorff

[gleaming warren]

thats fairly okay (for analysts)

[unreal stratus]

Best thing is when spaces don't even have a topology

e.g. simp sets

[gleaming warren]

Just discrete topology

that's also hausdorff nice

[tribal palm]

trivally hausdorff

[cunning mural]

t/f?: every (not necessarily open) subset of the real numbers is a countable union of intervals

seems intuitively true but I can't find a proof online

[hexed steppe]

why does it sound intuitively true

[cunning mural]

I take it back. the more I think about it the more I don't believe it

hmm I think I have an example

[hexed steppe]

how about {0}

or a cantor set

[cunning mural]

oh like that's [0, 0]

[cunning mural]

oh right

[hexed steppe]

ok cantor set then

[cunning mural]

oh wait

just the irrational numbers

[hexed steppe]

any uncountable set with dense complement

[sullen nimbus]

How to do 12.4x using 12.5x

I was able to figure out 12.5x

But i cannot phrase 12.4x in terms of 12.5x',s statement

What is the required property i should be looking for?

[simple musk]

commuting with an element of H?

[sullen nimbus]

Just an element?

[simple musk]

yeah fix h

[sullen nimbus]

Okay then why is there a neighborhood of that element that satisfies this

[hexed steppe]

what a bizarre problem

[sullen nimbus]

I guess I am supposed to use xgx^-1 is cts

[hexed steppe]

isnt this like 1 line using the definition of continuity

without this induction thing

[sullen nimbus]

Yes

[sullen nimbus]

I don't see how

Consider f(x)=xhx^-1 ?

[simple musk]

yes

[sullen nimbus]

Uhhh

Okay somehow I make use to the fact that H is normal, and H has discrete topology now?

How tho

Okay xhx^-1 is completely contained in H?

[simple musk]

yes

[sullen nimbus]

And f(h)=h?

So {h} is the required neighborhood?

No

I guess not

[simple musk]

maybe i led you astray. the usual thing would just be to look at the preimages of the points of H under f. theyre all open and disjoint so there has to be only one since X is connected. the hint is confusing me though

how about the property is "f is constant on the subset"

[sullen nimbus]

Uhhh

Yes this could work

I guess

[tribal palm]

P(R) is an incredibly complicated object…

even just R is

[tribal palm]

all the topologies i’ve seen given as examples of disconnected spaces, have been subspaces… are there examples for which this is not the case?

oh a discrete topology

works

bit boring though :S

[alpine nest]

How about the space of all binary sequences with product topology (with discrete topology on each copy of {0,1})?

Although that's cheating, since that's just ||the Cantor set||

[rapid lagoon]

Wait u mean the set that has the cardinality of [0,1] but (Lebesgue) measure 0?

[white oxide]

You mentioned one yourself earlier, GL(n)

[tribal palm]

oh, right

[tribal palm]

wait no this is disconnected yes but it has the subspace topology from R^(n x n)

[alpine nest]

It doesn't have to have Lebesgue measure 0

[ebon galleon]

Well I mean yes a lot of the spaces people are interested in embed into R^n

[opaque scroll]

The p-adic numbers is disconnected, and isn't usually thought of as a subspace of any other space. (Of course you can always embed a disconnected space into a connected one by just adding one more point)

[white oxide]

I mean, this seems a bit arbitrary though? As jagr explained you can always embed something into a connected space

[opaque scroll]

I guess it comes down to how you usually define / think about the topology.

The topology on GLn is usually given as a subspace topology

[unreal stratus]

Well there is a difference in terms of where the space comes from ig

[white oxide]

What was the AG examples you were mentioning? Are they more interesting than spectrum of product situations?

[unreal stratus]

AG was almost the opposite of what I want since a lot of stuff is assumed irreducible ig lol

So I delet

[white oxide]

Delet

[unreal stratus]

I guess there are lots of very natural examples coming from maps. Like if you consider the space of continous maps M -> M for most spaces lol

But then GL_n is very close to that

[opaque scroll]

Let R be the subring of (Z/2)^N of eventually constant sequences. Then the spectrum of R should be homeomorphic to the one point compactification of N.

Don't know if that counts as an interesting example.

[opaque cloud]

I saw "spectrum of R" and got real confused for a moment, until I saw that R is a ring

[tribal palm]

right this is a very nice example

thank you

[tribal palm]

god this is probably to some extent due to me currently being sick and not eaten properly this week (i have < 1$ in my savings rn and i have to rely on others for food ehehe), but i find proving these simple statements about connected spaces way too difficult 😭

[alpine nest]

The thing that gets me about connectedness is that it's such an intuitively obvious concept, but so tricky to deal with rigorously.

See also, dimension (unless we're talking linear algebra)

[tribal palm]

i will simply skip most of the results directly about connectedness for now… and hope i’ll have time to return to it before the exam (unlikely)

[tribal palm]

i will just jot down X connected and f: X -> Y contiunous implies Y connected, as this seems immensely useful mwahaha

[pallid delta]

it is
All things preserved by continuous functions are important

also path-connected -> connected is mega useful

[tribal palm]

i haven't even glanced over path-connectedness yet haha

though i have some familiarity with it in the case of complex analysis

[alpine nest]

Yeah, "is a continuous image of a known connected space" and "is path-connected" are usually the most convenient ways of proving connectedness.

And for disconnectedness I think I'd plump for finding a continuous function from the set into {0,1}

[pallid delta]

which is a funny way of saying a partition into two open sets, but it works

[alpine nest]

yep

I think a lot of people find it easier to think in terms of functions rather than in terms of sets

even if it's actually an equivalent formulation

[pallid delta]

I'd prefer seeing {(x, y) in R² | xy != 0} as partitioned into 4 sets though, for instance
It feels more natural since I don't add a layer of having a function, and checking it's continuous, on top of that

[alpine nest]

yeah, if you can just easily apply the definition, that's the best

[tribal palm]

oh that's good to hear

[alpine nest]

not right now, I'm on the phone and it's too much writing

it's just that both of those statements imply connectedness

and tend to be most convenient to verify in practice

[pallid delta]

they're just the two most useful ways of showing something is connected, mainly because they let you not think about what it means for that space to be connected

[tribal palm]

you can show that Y is connected by constructing a continuous function from some connected set onto Y

[pseudo coral]

as cont preserves connectedness ^^

right

[pseudo coral]

only if f is onto.. otherwise you just have that f(X) is ocnnected in Y

[tribal palm]

oh, uh, sure

[pseudo coral]

go ahead

[tribal palm]

i love it when i get to finish my proofs with exclamation marks mwahaha

[safe torrent]

5!

[pseudo coral]

thats not very professional lol

[tribal palm]

i don’t care mwahaha

[safe torrent]

Where did the square go

[pseudo coral]

how many math texts do you see that end a proof with ! lol

blacksquare )

[safe torrent]

mfw when \square is not used

[tribal palm]

i use \qed

[safe torrent]

this one is acceptable

[pseudo coral]

yeah either \qed or \blacksquare

[safe torrent]

But exclamation

[pseudo coral]

both work I believe i use the latter at end of my proofs

[tribal palm]

i mean, i’m not saying i use it as a end-of-proof marker

[safe torrent]

Use as an end-of-proof marker

[quiet thorn]

Use as an end of proof marker

[pseudo coral]

i may be new to this but did yall know theres a topological proof for infinitude or primes ?

[tribal palm]

i just mark my excitement in my sentence explaining the contradiction mwahaha

[safe torrent]

One step closer to crankery

[tribal palm]

imagine not letting people have fun with their proofs

[rapid lagoon]

It's attributed to furstenberg i believe

[tribal palm]

couldn’t be my prof

[pseudo coral]

yep it sure is I just read it now SO clever

[tribal palm]

rings a bell

[rapid lagoon]

:)

[pseudo coral]

its sooooooo slick

i mean i thought Euclids was sick when he introduced the product of primes plus 1

[rapid lagoon]

I will read it one day

[pseudo coral]

its a very clever technique

[tribal palm]

i believe some top exam here had a problem with a topology in which the open sets were subsets of primes or something

[alpine nest]

fun? In my mathematics?

[pseudo coral]

https://en.wikipedia.org/wiki/Furstenberg's_proof_of_the_infinitude_of_primes#:~:text=In mathematics%2C particularly in number theory%2C Hillel Furstenberg,a statement about certain properties of arithmetic sequences.

[tribal palm]

i’m taking my top exam late this year, but the ones who already did were given a problem in which you had to come up with a proof closely following urysohn’s metrization theorem… i am worried

[pseudo coral]

@tribal palm you eant a list of top exercises with solutions

want**

[tribal palm]

sure why not i can glance through it

[pseudo coral]

ill message it to ya!

@tribal palm

sent 🙂

if any1 else wants it I can drop it here too?

[tribal palm]

wonderful, thank you!

[pseudo coral]

or by messagw request only i say

[pseudo coral]

youre welcome! I also have one for complex and one for measure

they have hyperlinks in them which sends u from contents to a problem and back to contents @tribal palm

[alpine nest]

You need to rigorously define what "connected to each other" means

Since you're using the term "fractal", you presumably mean some kind of self-similarity

[ebon galleon]

w h a t

[alpine nest]

So you should look into rigorous definitions of self-similarity and try to apply one of those

If your aim is to make profit out of this, I recommend either writing up some persuasive bullshit and selling it, or making contacts and doing insider trading

Those are the most reliable ways to make money out of markets

[hexed steppe]

brownian motion has a fractal structure

if that is what you are asking

do you smoke fentanyl while day trading

thats probably a good sign

is the 1488 in your username also part of your trading strategy?

like a public key or something

[white oxide]

Sure

[hexed steppe]

i think if that was true you would have found my question too weird to answer

[rapid lagoon]

fractal fentanyl transform?

Yeah aint no way ur not a troll

"vortex methematic" lol

[hexed steppe]

lets come back to the 1488

[quiet thorn]

how about you take this conversation to #chill or #discussion

or at least localize it to one advanced thread or something

[white oxide]

who are we formally inverting?

[tribal palm]

uhhh something is wrong with the statement in 12.C here right

[hexed steppe]

like what