point-set-topology

[round flint]

converges in T_1

I forgot to add

[fervent citrus]

this what you wanted to do, has your goal changed ?

[round flint]

(x_n) converges to irrational number

[wanton marsh]

then what is an integer m such that n >= m implies 1/n in {0} ?

[round flint]

Sorry I forgot to mention

[wanton marsh]

if you think it converges in T1 then you are saying that such an m exists

[round flint]

yeah you're right

it doesn't converge if x is rational

[wanton marsh]

yeah unless the sequence is eventually constant

[round flint]

yes

[wanton marsh]

so then if you assume the limit is irrational

so all you need to do is to show that any basis open for T1 that contains the limit also contains a standard open containing the limit ?

[round flint]

Right

[fervent citrus]

isn't this upside down ?

o wait nvm

[round flint]

No?

Then if it converges in T it would converge in T_1

[fervent citrus]

i had a brain lag

[round flint]

And it seems trivial

Because for x irrational contained in [a, b], a < b

then since $(a, b) \subset [a, b]$

[gentle osprey]

Mr. M:

[round flint]

and open neighborhood is a union of basis elements

so for any open neighborhood in T_1 there should be a smaller one in T

[wicked mirage]

show that every uncountable closed subset of a complete separable metric space contains a subset homeomorphic to the Cantor set

the problem lies in proving that there is a subset such that is both totally separated and compact

(and is nonempty)

hmm

[static hinge]

Probably stupid question, but assuming I have a normal cover P\to X with group G, what is Aut_X(P)?

[gentle osprey]

slimvesus:

slimvesus:

[static hinge]

So far so good, maybe I should elaborate a bit where my confusion comes from, so they say principal(normal) G-bundle P\to X, so I have a G Action that is free and transitive on fibers, meaning it should be Aut(P) itself right?

And like they r looking at all of those and they Aut() is invariant under isos respecting X, but what’s the point to even look at Aut if it’s just G on any iso class

Or am I just taking something wrong here?

I’m not totally sure if I’m honest

https://web.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf

These lecture notes

Page 17 is where my confusion lies

Where they define the measure as 1/Aut

And the fields itself are just principal bundles with group G, but like if all of those have the same Aut group, what’s the point of defining a measure like this...

[gentle osprey]

slimvesus:

[static hinge]

We can always just take the base to be a point anyway and just have arbitrary groups

But for a point any cover is Normal anyway

Do covers exist that are G-bundles but not normal?

That could possibly explain this, and I’ll just ignore the normal Galois word in those brackets lol

I mean he starts the page with let G be a group

He also says later on that it’s finite because it’s just hom(pi1,G)/G

So it’s definitely a fixed group

[gentle osprey]

slimvesus:

[static hinge]

Umm what is this C for?

Anyway thanks for thinking about this 🙂

[clever lantern]

hi guys, i have some problems to find a way to solve the following question: Show the map $f : R^2 \to ; [0,\infty) \times R$ is an open map with $f(x,y)=(|x|,y)$

[gentle osprey]

Black:

[fervent citrus]

what have you tried ?

[clever lantern]

i have tried to see how the open balls $D(\epsilon,x)$ of $R^2$ behave when mapped by f and if $D(\epsilon,x)$ are a subset of $R^2 | x>0$ the map behave like identity

[gentle osprey]

Black:

[clever lantern]

and i've come to a dead path because i don't how to show a generic $D(\epsilon,x)$ in open in $[0,\infty) \times R$

[gentle osprey]

Black:

[clever lantern]

generic $D(\epsilon,x)$ with $x>0$

[gentle osprey]

Black:

[clever lantern]

before this I tried to see if I could rewrite an open ball as a product of open intervals

[uncut surge]

If you struggle with round balls, you could also try using open squares instead; remember that the topologies generated by round open balls and open squares are the same in the real plane

They seem better adapted for this precise problem

[clever lantern]

@uncut surge yes I know, in fact from there I thought to rewrite an open ball as a product of open so as to have an open square; so you're suggesting you build a homeomorphism between the squares and disks before evaluating the map?

[uncut surge]

That sounds a bit more complicated than I was thinking, actually... No, what I'd suggest is that you show that if you have an open square A, then f(A) is open

That then automatically shows that if you have any open set U, its image f(U) is open (because squares and balls generated the same topology)

If you don't know that squares and balls generate the same topology, you might wanna think about that statement a bit

[clever lantern]

because each open ball can be written as a union of squares and therefore the family of squares is also a basis for the Euclidean topology

[sweet wing]

Anyone knows of some example of a contractible space as a subset of R^2 that is not pointed contractible to any points(i.e. any point in the space is not a deformation retract)

[sweet wing]

Some spaces are contractible but not deformation retract at some point

For example the comb space, if you set your point at like (0,1) or (0,0) depending on how you define the comb it is not deformation retract there

no point

hm

am doing through the more annoying exercises in tom dieck at in 2.2 exr 4 he seems to suggest such a subset of R^2 exists

K deformation retract of X if theres some homotopy from X to K that doesnt move K i believe

yea

ahh yea

feels strange for a contractible space to have such a property

[coral pivot]

i mean isnt the comb space a subset of R^2?

[sweet wing]

yes

but it is (strong) deformation retract as some points

[coral pivot]

oh hmm wikipidea says it doesnt

[rugged swan]

for me the definition of contractible is that the inclusion pt->X is an homotopy equivalence, so it is the definition of a retract isn't it ?

[sweet wing]

as long as it isnt like the (0,0) or (0,1) point(depending on definition)

retract here is stronger, the homotopy cant move pt

where pt is treated as a subset of X

[coral pivot]

yeah its cause you are working in htpy* instead of just htpy

[sweet wing]

yup

Basically every point including origin except on the y axis

yea

needs some cursed space that acts like the y axis of the comb space but without the other points

[coral pivot]

yeah hmmst

i think hatcher actually had an example

hold on

yeah it did some cursed zigzag space

let me post screenshot

oof

lol i was about to delete too

[sweet wing]

ok yup that's cursed but works

[sleek thicket]

Let X be the subspace of C^n of all tuples (z1,...,zn) where zi ≠ zj for i ≠ j. Let Y be the set of all size n subsets of C. There's an obvious surjection p : X -> Y, and we can topologize Y by declaring that p is a covering map. Clearly each fiber of p has cardinality n!. Is p a covering map? I think it is, since if you have a point S = {z1,...,zn} in Y, you can let ε = min{|zi - zj| : i ≠ j} and the product of all disks of radius ε/2 should evenly cover a neighborhood of S

Like let Ui be the disk of radius ε/2 about zi. Then let V = p(U1 × … × Un). I think V should be an evenly covered neighborhood of S, with p^(-1)(V) being a disjoint union of the connected open sets U_σ = Uσ(1) × … × Uσ(n) for σ in Sn, and each of U_σ should be mapped homeomorphically onto V by p

Err I guess I need to intersect U_σ with X, since it contains points with duplicated coordinates. I still think this should go through though

[marsh forge]

Sorry could you explain your obvious surj

Oh I see you just forget the ordering

Right

[sleek thicket]

Oh yeah sorry

Obvious means "i've been thinking about it for a while"

[strange folio]

We need a word for "I thought about this for a while and it was clear to me, and I don't think I can say anything to make it faster for you, but I'm confident if you think about it for a while it will be clear to you"

[sleek thicket]

It would probably have been clearer if I defined an equivalence relation on X explicitly and let Y be the quotient

[honest narwhal]

Ugh this suggests Max is immune to my syndrome

For me the word obvious is "I want to make it such that people feel too guilty to ask"

[strange folio]

the day before my oral exam I went through my paper and searched for every time I used the word "clear" or "obvious" to make sure I could quickly explain why it was true in case they asked

I think the best way to dodge people asking is to use the word "routine"

because that makes it sound like you're sparing them something boring

"The proposition is a routine application of the Grothendeik-Ogg-Shafarevich formula."

[marsh forge]

I embrace my stupidity

Its much more fun to not know something bc its fun ti learn

[dire warren]

Max ur profile pic

Doesn’t look like u

Why?

It’s usually u

[honest narwhal]

Doesn't it?

Last time I met him irl that's about what he looked like

[dire warren]

I mean there is some resemblance true

[coarse kestrel]

[marsh forge]

My friend painted it for me for my birthday

[supple locust]

tensor field is section of a cotangent bundle right?

[tight agate]

a 1-form is a section of the cotangent bundle

[supple locust]

yes i get that

what is tensor field then if not 1 form?

oh is it just any k form?

[tight agate]

a k-form is a section of the kth exterior power of the cotangent bundle. iirc a tensor field (at least in the context of manifolds) is a section of a bundle that looks like a tensor product of a bunch of tangent and cotangent bundles

so something like $TM \otimes ... \otimes TM \otimes T^*M \otimes ... \otimes T^*M$

[gentle osprey]

Brofibration:

[supple locust]

i have a silly question

what do we do of these tensors/tensor fields

[tight agate]

Have you seen vector bundles in action before?

[supple locust]

yes

not exactly sure what do you mean by 'in action' tho

[gritty widget]

if you're asking about how to think about them the answer is serre-swan

https://ncatlab.org/nlab/show/smooth+Serre-Swan+theorem

[supple locust]

ill check the link

[gritty widget]

hold up i'll find a more readable version

the statement on this page is fine https://ncatlab.org/nlab/show/Serre-Swan+theorem

[supple locust]

val, i know words in the article but it still seems very diffiult to understand
i am currently taking first course on differential geometry. ill take some time to understand it

[gritty widget]

oh i was reading above

$(k,l)$-tensor fields are defined to be sections of $$\bigotimes^k TM\otimes \bigotimes^l T^M=\bigcup_{x\in M} \bigotimes^k T_xM\otimes \bigotimes^l T^_xM$$

[gentle osprey]

val:

[supple locust]

yes i think i understand this definition

are tensor fields used in stokes thm ?
cause thats what coming up next

in my course*

[gritty widget]

strokes says something about the integration of forms on oriented manifolds

which are tensor fields yes

[supple locust]

@jaunty ferry do you know what basic open sets are?

@gritty widget yeah but is knowing forms enough?

so do you know topology generated by base?

[gritty widget]

just knowing the definition is not

read https://maths-people.anu.edu.au/~andrews/DG/DG_chap13.pdf or something first

[supple locust]

you take all possible unions

@gritty widget sure will thanks

not necessarily minimal

[gritty widget]

oh wait up sorry i made a typo

[supple locust]

@jaunty ferry base is collection of open sets such that any open set is union of open sets from this collection

@gritty widget i am in fact reading up on forms from Lee

i wanted to know if i have to know tensor fields in detail

[gritty widget]

actually no i did not lol

lee is good

[supple locust]

@jaunty ferry so do you know what is base for product and box topology ?

@gritty widget LOL

yeah but i am kind of lost in the details

check them on wiki @jaunty ferry

[gritty widget]

problem with differential geometry is all the indices

[supple locust]

@gritty widget i would say that is bookkeeping and can be managed. difficult part is making sense of abstract definitons like vector bundle. for me atleast

how are they defined ? if you can answer this question then you are done!

https://math.stackexchange.com/questions/871610/why-are-box-topology-and-product-topology-different-on-infinite-products-of-topo

[sweet wing]

yea it shouldn't happen
categorically product topology is a lot more natural and box topology

Just a quick sanity check

If $p:E\to B$ is shrinkable, then $(E,p)$ is homotopy equivalent to $(B,\text{id})$ in $\mathbf{Top}_B$, in other words:

There exists some maps in $\mathbf{Top}_B$, $f:E\to B$ and $g:B\to E$ such that there exists some homotopy $H_t$ with $H_0=g\circ f$ and $H_1=\text{id}_E$ with $p=p\circ H_t$ and $f\circ g=\text{id}_B$(let $G_t$ be a homotopy with $G_0=f\circ g$ and $G_1=\text{id}_B$ with $\text{id}_B=\text{id}_B\circ G_t$, so $G_t=\text{id}_B$)

[gentle osprey]

ariana:

[sweet wing]

it's ok right

[fading vale]

is shrinkable = contractible

@sweet wing

[sweet wing]

not rlly

[fading vale]

what is Top_B here?

[sweet wing]

contractible is space homotopic to a point i believe?

lemme open up tikzcd XD

[fading vale]

lol

yea ive just never seen the term shrinkable before?

is this a htpy thing or a point set open cover meme thingy

[sweet wing]

idk it is like a random thing i skipped previously in tom dieck XD

[fading vale]

tom dieck monka

i think this is like uhh

[sweet wing]

points in $\text{Top}_B$ are $(A,a)$ with $a:A\to B$ and morphisms between $(A,a)$ and $(C,c)$ are maps $f:A\to C$ such that $a=c\circ f$, so basically we get this triangle
\begin{tikzcd}
A \arrow[rd, "a"'] \arrow[rr, "f"] & & C \arrow[ld, "c"] \
& B &
\end{tikzcd}
so this is spaces over $B$

[gentle osprey]

ariana:

[fading vale]

uhhhh hm

whats a here

a continuous map in normal Top?

a htyp equivalence?

[sweet wing]

yee

[fading vale]

first one?

[sweet wing]

not necessirely htyp equiv

[fading vale]

ok hmm

this is very

[sweet wing]

agreed

[coral pivot]

hmm is g:B-> E a map in Top_B tho

[fading vale]

i guess maybe we should work thru this concretely and start but just choosing a B?

uh what would a good B be

RP^2? or smth?

[sweet wing]

ok a bit lazy when i typed $f:E\to B$ i meant like $f:(E,p)\to (B,\text{id})$

[gentle osprey]

ariana:

[sweet wing]

B={*} Top_B becomes Top hehe

[fading vale]

what are some examples of nonshrinkable spaces

non normal ones

i think

Zariski top on Spec(Z)

[sweet wing]

(shrinkable is in regards to the map $p:E\to B$ here)
let $E=B+B$ and $p$ be the obvious injection

[fading vale]

h u h

[gentle osprey]

ariana:

[fading vale]

how are u defining shrinkable then

[sweet wing]

If p:E->B is htpy equiv with id:B->B in Top_B, so (E,p) htpy equiv to (B,id)

[fading vale]

ok so is this ur defn of shrinkable?

do u want an example of a non-shrinkable space?

[sweet wing]

the injection from B+B to B isn't shrinkable

i just need a sanity check to make sure im sane lol

so like it should be eqiuvalent to saying

some maps f:E->B, g:B->E in Top_B with homotopy from gf to identity on E and fg is identity on B

[fading vale]

this is kind of wack

im not sure i understand the Top_B memery well enough to help

sorry D:

[sweet wing]

same myself

i have not enough exotic stuff to play with in Top_B XD

[coral pivot]

whats B+B here

[sweet wing]

$B\coprod B$ kek

[fading vale]

its very hard for me to conceptualize Top_B geometrically

[gentle osprey]

ariana:

[coral pivot]

oh smh

[fading vale]

i would start with fixing a B

[sweet wing]

yea i should probably not use + XD

[fading vale]

and trying to get examples of spaces in it

to develop some intuition

[sweet wing]

hm true

[fading vale]

what is this even used for

ive never seen this b4 and when i googled i found nothing

[sweet wing]

idk it appears basically nowhere

XD

[fading vale]

tom Dieck pls

[coral pivot]

lol

[sweet wing]

lol

[sweet wing]

so basically the counterexample im finding is some htpy equiv p:E\to B that has a section but is not shrinkable.

If we let E to be the space Y in https://media.discordapp.net/attachments/496785752936546304/718140633742573630/Screenshot_from_2020-06-04_09-34-19.png and B be the bolded line then p is a htpy equiv that has a section but you can't construct the H_t from E to E that satisfies the appropriate commutativity relations in Top_B

cursed spaces

[reef ether]

Counter Examples in Topology is a great book to flip through on occasion.

Pathological examples bolster the mathematical mind's immune system I suppose 😂

[marsh forge]

Counterexamples in topology should be burned

[sweet wing]

it is helpful isn't it

yea
i kinda want a counterexamples in algebra lol

[marsh forge]

Its a joke from my first alg top professor

He told us all to buy a copy and burn it

If you want to know why I actually agree w the sentiment its because as far as im concerned the pathological counterexamples show our defn of topology was too naive, not that these are real spaces anyone should care about

[ivory dragon]

as far as im concerned the pathological counterexamples show our defn of topology was too naive

what is implied by "naive" here

if you mean "broad", then yeah, thats the whole point

if you mean "bad"

then i strongly disagree

the point of a topological space is to be a really really general fucking object that becomes substantially less general when you add like

any structure

CiT, if anything, shows how "important" this added structure is

[marsh forge]

Uh maybe this is a usage of the word that isnt common beyond my dept?

Naive meaning like, oh this definition looked like it might be good enough to capture all the stuff we like and get rid of the stuff we dont

But really we wanted more conditions

[ivory dragon]

but i think "naive" tends to carry an implication of being "bad"

i would not consider the definition of a topological space bad/flawed

[marsh forge]

Like “naive” vs “genuine” is a common terminology in topology for old vs improved defns

Oh sure

I didnt really mean it that way

[ivory dragon]

the point of CiT is exactly to illustrate that:

the notion of a topological space is really general

so you need some additional definition on top of it

before you can make "natural" statements

in full confidence

[marsh forge]

Yeah I dont have anything against the book/author

[ivory dragon]

i think it achieves that goal fairly well

while also being, you know

kinda neat

[marsh forge]

But i do think ppl fetishize those counterexamples in a way i think is kinda silly

[ivory dragon]

even if the spaces are esoteric and generally not the most useful things to consider

[marsh forge]

Like ooooo the l o n g l i n e

[ivory dragon]

do people do that?

i havent really seen that

[marsh forge]

Yes

Its mostly young ppl

[ivory dragon]

well i mean

at least they're falling for the mysticism kool-aid in general topology instead of cat theory or some shit

¯_(ツ)_/¯

[marsh forge]

Idk if i think thats better lmaooo

[ivory dragon]

i mean like, theres a certain sense that we only care about CiT up to how much we care about general topology

and theres not much reason necessarily to care about general topology unless youre like, a logician

but idk, if people find it neat

nothing wrong with that

idk i dont know any logic

[marsh forge]

Set theory

[ivory dragon]

i know topology is relevant to logic and the spaces used generally have less structure than they do in AT or whatever

[marsh forge]

I think they add different conditions

Making the general sefn still useful

Defn*

[ivory dragon]

i dont know exactly how much "less" is

because again

not too experienced

with logic

but i mean if you wanna criticize anything about the "fetishization" of topology maax

i'd criticize the undergrad curriculum seeing a full semester of gen top as "core"

in the same regard as ring theory is core

[marsh forge]

wait what

[ivory dragon]

like realistically most undergrads only need the first month of a gen top course

[marsh forge]

I agree that gen top is too long

I didnt even bother to take it

But ring theory?

Id extend it if anything

[ivory dragon]

no im saying

[marsh forge]

Ok but plenty of mathematicians use things every day that arent core jan

[ivory dragon]

the fact that general topology is regarded as a "core" course on the same regard as ring theory

[marsh forge]

Oh

Sure

[ivory dragon]

in that its a prereq for fucking everything

[marsh forge]

Idk

[ivory dragon]

feels a bit wrong

[marsh forge]

Topology should be there imo

[ivory dragon]

like general topology is important for some people

[marsh forge]

Yeah idk

[ivory dragon]

mine?

[marsh forge]

I think that top is as important as like

[ivory dragon]

i dont think ive ever used the second half of munkres

yes

[marsh forge]

Oh sure

[ivory dragon]

but i only need like the first few chapters

i mean idk

maybe i like

use the intuition from later chapters of munkres

[marsh forge]

I feel like theres two diff args here

[ivory dragon]

¯_(ツ)_/¯

[marsh forge]

The first is that the first few chapters of eg munkers is necessary

I agree

[ivory dragon]

yeah theres no argument about that

that needs to be taught at some point

[marsh forge]

The second is that the rest of gen top is abysmal and useless

Which i also agree

[ivory dragon]

and general topology should be offered as

an elective

since it is relevant

[marsh forge]

It is

At uchi

[ivory dragon]

for some mathematicians

[marsh forge]

I do think its sad that one can graduate undergrad without even seeing a defn of topology tho

While Galois theory is trumped up like its as important as ring theory

Honestly galois theory as a standard course needs to be electivized

[ivory dragon]

ok i strongly disagree there max

i think everyone should be familiar with galois stuff

[marsh forge]

Why

Should they spend 10 weeks on it?

Yes

Hence 10 weeks

[ivory dragon]

i mean it just feels like part of a "mathematician's repertoire" in the sense that

i feel like you cant even understand the like

motivations or processes

of, say, alg NT

without galois stuff

[marsh forge]

Yes

Alg NT people

Should learn galois theory

But its not nearly universal enough

[strange folio]

Galois theory is also a model for a ton of stuff that happens in other areas, like Algebraic Topology

[marsh forge]

I disagree really

What example in AT

Do you have in mind?

Covering theory?

[strange folio]

Covering spaces are a nearly exact analogy.

[ivory dragon]

and honestly i feel like galois theory teaches

[marsh forge]

Yeah but that analogy is not strong unless you degenerate to riemanniam geo

[ivory dragon]

"algebraic thinking"

very well

like ok im gonna be frank here

most of an intro group/ring theory class

is definition pushing

like basically all of it

[marsh forge]

I also think those classes should be modified for sure

[ivory dragon]

galois theory is a nice "middle ground" between that and actually having novel ideas

[marsh forge]

But like idk I could have learned all the galois theory id ever need

In 3-5 weeks

Rather than 10

[ivory dragon]

ehhh

i guess thats maybe fair

but how do you structure an undergrad curriculum such that

the 40% of undergrads that need it have a class for it

but the remaining 60% only cover half of it

[marsh forge]

Shorten galois theory add soem other stuff

[ivory dragon]

yeah but then when will they learn galois theory?

[marsh forge]

Wdym

At the end of the alg sequence

[ivory dragon]

no im saying like

if you drop half of galois theory

then you have a "rump" of "leftovers"

that a lot of undergrads need to know

but that doesnt really justify

a full course

i guess mayyyybe you could have like a

"galois theory and algebraic number theory" course

[marsh forge]

Have an advanced galois theory course

Yeah

[ivory dragon]

at an undergraduate level

[marsh forge]

Again I feel like the core requirements

Should not be built

To supplement niche interests

[ivory dragon]

but i would not consider GT niche lmao

yeah its not necessary for everyone

[marsh forge]

Like you can graduate from uchicago without knowing what cohomology is

Or a category

[ivory dragon]

but its necessary for a lot

[marsh forge]

And id say those are way more useful

Than galois theory

[strange folio]

I mean certainly if you're going to study PDEs or something, you don't need to take Galois theory (though its still beautiful). But otherwise, the ideas from it are pretty essential, even if you're not directly applying them.

[ivory dragon]

tell that to analysts?

like is CT useful at all in analysis

[marsh forge]

I could make the exact samr arg nami

About galois theory

And every other subject

[ivory dragon]

sure

so if youre argument is you should make GT not-core

[marsh forge]

Category theory is the bread and butter of like, a huge chunk of math

[ivory dragon]

i'd argue the same for CT and stuff

[marsh forge]

Would you say thats less fundamental

Than galois theory? Really?

I agree

[strange folio]

I mean, Galois Theory is going to be part of a grad algebra course anyway, so I don't think it is obligatory for undergrads.

[ivory dragon]

my grad algebra course assumed you knew GT

[strange folio]

but there is no way an undergrad is going to get all the math that they technically "should" know for grad school.

[marsh forge]

I think the big ideas and findamental theorem could be beelined quickly

[ivory dragon]

admittedly it probably wouldnt be hard for a motivated student to self-study

but still

[marsh forge]

Spending weeks computing galois groups of polynomials no one cares about

Is objectivrly a waste

For most ugs

[strange folio]

Haha, that hurts me

[marsh forge]

Sorry ahah but topologists dont make every ug compute 100 fundamental groups

[ivory dragon]

i mean

my galois theory course didnt do that?

[marsh forge]

Mine ran out of theory in like

5 weeks tops

[strange folio]

I think computing Galois groups of polynomials is the first time you really see number theory. But I don't think undergrads have to do it. But I definitely think grad students have to compute 100 fundamental groups

[marsh forge]

We deep dove into like, algos for polys of specific degrees

W determinants

And resolvents

Tbh i wanted to commit sepuku

[strange folio]

yeah, I don't think that stuff is super worthwhile

[ivory dragon]

yeah okay my course didnt do that so like

mmaybe i just havent seen this course

[strange folio]

but the fact that, e.g., x^4+2 and x^4+3 are fundamentally different even though they shoudl be "the same" is important

[ivory dragon]

youre referring to

[marsh forge]

It might he the perfect storm

Of frank caligari and matt emerton

Creating the worlds worst algebra class

[ivory dragon]

we did do some

[marsh forge]

Bc frank is obsessed with computations (sometimes rightly so)

[ivory dragon]

ruler-and-compass shit

that no one cared about

[strange folio]

but I would say, resolvants are a bad bad thing to emphasize in a course because they are A) not useful that often and B) easy to learn.

[marsh forge]

And Matt is obsessed w going slowly

[ivory dragon]

including the lecturer

[marsh forge]

Oh god

I didnt even read that chapter

I figured id take the bullet on that pset

I might tear this class apart in the review

We spent like 2.5 weeks on 14.6 in dummit

[strange folio]

I think the high level summary of the ruler/compass stuff is really cool. "Solving quadratic equations only gives you degree 2 stuff, so everything you get lives in quadratic extensions of quadratic extensions of quadratic extensions of..." so you definitely don't get cube root of 2.

[marsh forge]

And every pset was just miserable

Idk my problem w this is like

Theres a lot of stuff i find really cool

[ivory dragon]

oh yeah and the ruler-compass stuff is historically very very relevant

[marsh forge]

I think thats a terrible metric

For whether something is a good topic

[strange folio]

but I struggle to spend more than half a lecture on the ruler/compass stuff.

[ivory dragon]

in that it shows one of the most direct applications of the more abstract framing of algebra

but yeah

we spent like

[marsh forge]

Bc it changes hugely from person to person

[ivory dragon]

a good 2 weeks on it

and it was miserable

it really didnt warrant that much time

[marsh forge]

Like i could not care any less about ruler compass stuff

[ivory dragon]

meanwhile we talked about diagonalization and canonical forms and change of basis matrices and the gram-schmidt process

all in the same lecture

of linear algebra

[strange folio]

yeah, the point, in my mind, is "From this point of view, this historically important and difficult problem is completely trivial"

[ivory dragon]

so like

lmao

[strange folio]

if you spend two weeks on it, you really undercut the "is completely trivial" part 😛

[marsh forge]

Also i dont think historical relevancy is good either

[ivory dragon]

i mean i think its worth mentioning

[marsh forge]

No one i talked to found “quintic solving” to be a good motivator

Like

No one cares about solving quintics

[ivory dragon]

i care

😠

[marsh forge]

Lol sorry

[strange folio]

I'm certainly a mathematician because for reasons I cannot explain, my high school taught a math history course and I found it fascinating. I did my final project on quintic solving.

[marsh forge]

I have bad news 4 u

Yeah I mean Im not surprised some peoole find it interestint

Maybe jan

I cant argue that

[ivory dragon]

oh i wouldnt say i find galois theory all that interesting

but i think its important for a "well-rounded" mathematician just to be familiar with the "basic repertoire"

i wouldnt mind having them spend less time on it

[strange folio]

But I'm huge on teachers communicating broad connections to history and other areas. That's what teachers can do well, that textbooks cant' do so well.

[marsh forge]

I dont think it matters whether its interesting or not. I dont find analysis interesting but i dont disagree it should he core

[ivory dragon]

but i think everyone should at least have like a

[marsh forge]

Be*

[ivory dragon]

surface-level understanding

[marsh forge]

Surface level understanding is like 3 weeks nami

Yeah yeah ik

Idk if youre right

Ive never interscted w people that care about polynomials that much

Hell dummit has a chapter on AG

Thats a way more interesting segway

Lmaooo

Hey we actually have an LA requirement now

Ill have u know

Idk if you did like

Rings -> mods -> baby galois -> varietirs

I think thats way more interesting

Varieties*

Mine had 10

It was

Pain

Its over now

Thank fucking fod

God

Literally 3 weeks on the “Galois Groups of Polynomials” chapter

I have no idea how it was justified bc i didnt go to lecture even once

But the psets were awful

[ivory dragon]

Rings -> mods -> baby galois -> varietirs

is this not what everyone does

[marsh forge]

No

[ivory dragon]

i mean ok we did something like

[marsh forge]

10 weeks of galois and no more algebra nami

[ivory dragon]

groups -> rings -> modules -> BRANCH

[marsh forge]

I am done w algebra as far as uchi is concerned

[ivory dragon]

and from the branch there was

galois -> alg nt

and also

varieties -> comm alg -> schemes

[marsh forge]

1 year

1/3 of it is galois theory

Its cancer

[strange folio]

I mean, Galois Groups of Polynomials over Q is an incredibly important area of active research in number theory. It's one of the only things you can touch on in abstract algebra that connects to current active research.

[marsh forge]

Sure

But 1/3 of ug algebra

Being about it

Is insane

Imo

You could throw all sorts of stuff into a core curriculum

And justify it w proximity to research

I dont think thats a good idea

Hell throw spectral sequences into the algebra core for that lol

[strange folio]

Where I teach now, we have a semester of algebra, followed by independent study if a student is interested. Where I taught previously, it was trimesters, Groups, Rings, then Fields/Galois, which felt right.

[marsh forge]

Tbh given that this channel is dominated by regulars

[ivory dragon]

we had 4 semester of algebra but i mean that's including

intro to AG and comm alg

[marsh forge]

I think its okay for this to be our #general lol

Unless someone has an actual topology questikn

[ivory dragon]

well actually

we had 5 semesters of undergrad algebra

[marsh forge]

AG is like taught every other year here and has galois as a prereq iirc

[strange folio]

Well, when you get to Galois Theory for infinite extensions, and you connect it to topological groups, that's pretty sweet.

[ivory dragon]

but one of the courses was:

Group actions, Sylow Theory, solvable and nilpotent groups, Galois Theory.

and another course was:

Field extensions. Groups of automorphisms of fields. Galois theory. Finite fields and applications. Solvable groups, the insolvability of the quintic equation. Ruler and compass construction.

now you might tell yourself

"both of those are galois theory courses"

the answer is

"yes"

i honestly dont understand the motivation behind this structure

at all

but whatever

[strange folio]

Haha I was just connecting Galois Theory to the topology channel.

I think the finite group theory curriculum is overly influenced by what was important when finite group theorists were working on proving the classification theorem.

but curriculum change is always slow.

[marsh forge]

Idk abt that

Well

Actually yes thats true but like

[ivory dragon]

if only the proof of classification was actually written down somewhere

[marsh forge]

Z/n is my only real counter example

[ivory dragon]

rather than being spread in like 12 different sources combined with the folklore of like 100 finite group theorists

[marsh forge]

Finite abelian groups and infinite groups are my only friends

Well once some brave soul

[strange folio]

I would say that the most important part of finite group theory is emphasizing all the tools of studying group actions, because groups exist to act on things.

[marsh forge]

Writes it all down in coq

Youll have it nami

Lmfao

Group actions were under emphasized in my course for sure

[ivory dragon]

i dont think anyone actually knows the full proof

[marsh forge]

And my prof explicitly emphasized them more than the others

[ivory dragon]

they just know one subsection of the proof

stitched together with their own internal folklore

and believe all their peers have their own areas covered

[marsh forge]

All group actions are continuous

[ivory dragon]

real take:

groups exist to make rings out of

all other applications are vestigial

[marsh forge]

Groups exist to plug into cohomology

Jk if you use coeffs other than Z/p ur evil

[strange folio]

Generally, when I have taught graduate algebra, the majority of my time in group theory has been spent on group actions.

[marsh forge]

I thought you normally have to generalize out of Z/p to do equiv

Like in equiv topology

We use mackey functors

And mackey functor fields

Which are cursed but whatever

Can someone explain coends to me like im a stupid undergrad

Bc im a stupid undergrad

Coends are one of those things that when i saw it i was like

I can def blackbox this

And it has gotten to the point

Where i cannot do that

Oh boy

Who is faye

Fadingmidnight?

[fading vale]

does fadingmidnight know coends

[marsh forge]

Oh i should ask them then

[fading vale]

faye is good

[marsh forge]

I think mniip knows them but tbh i think we think so differently that it might not be helpful lol

[fading vale]

faye told me the negi umich math dept guy story which is very close to my heart

[floral gust]

I would say that the most important part of finite group theory is emphasizing all the tools of studying group actions, because groups exist to act on things.
"Groups, as men, will be known by their actions".

[slow gorge]

hello there, oddly enough, i am looking for a problem... i was once asked this question of : if you take a sheet of paper, take a second one, crumble it without cutting or puncturing it, and have the total 2d span of this ball of paper be inside the domain of the original piece of paper, show that there is bound to be a point who's coordinates on both pieces of paper are the same. maybe ive phrased it incorrectly, i was hoping somebody who was familiar with this problem could link me the proper wording! (and maybe even the proof , this is purely out of curiosity, this is beyond my level atm haha)

[bitter yoke]

This sounds like the Banach fixed point theorem

[slow gorge]

interesting, i think it is this yes

thanks!

[slow gorge]

hmmm... is the mapping i described a contraction mapping though?

cos mine allow for points to be superposed upon one another

i think it sounds more like Brouwer's fixed point theorem

Banach's seems to be some derivation of Brouwer's? yet the former was proven after the latter, which seems to suggest elsewise

[midnight jewel]

the two are not really related. in your case, banach’s isn’t applicable but brouwer’s is

banach requires a complete metric space X and a lipschitz-contraction, brouwer requires a convex compact set in ℝⁿ but then the function just has to be continuous

(more generally for brouwer it can also be convex and compact in a banach space, but that generalization is apparently known as “Schauder’s fixed point theorem”, I wasn’t aware of it)

[slow gorge]

isnt contraction continuous though? im wrong but thats how i understand it

also, topology is so interesting wtf, for the past hour ive been stumbling upon numerous interesting results

gotta wait like a year before we get to topology tho haha

also, is there a way to find this point?

[quiet mauve]

Forgive me but isn't the second condition redundant? Basis having all 'x's means that everything is already included no?

[chrome dew]

just because we have all the points doesn't mean we have all the sets of the topology

go back and check the axioms a topology needs to satisfy

[quiet mauve]

Ah. So if it wasn't a topological basis but just a basis it wouldn't need the second condition?

[chrome dew]

no such thing as "just a basis" here, the basis is for a topology

we care more about open sets than just points here

[marsh forge]

@slow gorge

Fixed point is found via iteration of the function

For contractions

[slow gorge]

mhm

[gritty widget]

try taking a point in the closure and looking at sequences in S converging to it

[marsh forge]

should that first function be S->R

[gritty widget]

bot pls

[gentle osprey]

Token:

[gritty widget]

yes, this is either the definition of closure or an easy equivalent one

metric spaces are nice when it comes to sequences

well, you can start by showing that if a point lies in the closure, you have a sequence in the original set converging to it

that should get you started

then try using that to define the extension of f

(maybe i gave too much away)

as long as you're careful about it, sure

also im not too sure about that definition of closure

ok, good

that's the harder part

[floral gust]

Would it be correct to say that the subspace topology is the coarsest topology that makes the retract continuous?

[strange folio]

Yes, that sounds equivalent to the definition about intersecting the open sets with X to me.

[floral gust]

Oh thanks. Any idea how to deduce the coarsest-ness?

[strange folio]

Well, let's say we have the inclusion X to our ambient space. Then the inverse image of an open set U in the ambient space under that map is exactly U intersect X

so that has to be in there

but the subspace topology is nothing else but those

so itis the coarsest

[floral gust]

Hmm, yeah I was aware about the subspace topology being coarsest for the inclusion map but wasn't sure about the retraction map.

Although the subspace topology gives us that the retraction is continuous, but I am not sure whether it is coarsest for it

r: X → A where A ⊂ X such that r restricted to A is the identity on A

[strange folio]

oh sorry, I thought you were using retraction to mean pullback!

[floral gust]

Oh right, it is useless to talk about coarsest for a map where the topology on the codomain is arbitrary.

Hmm right right

[floral gust]

@gritty widget But it talks about quotient topology from π : U → π(U) instead of π : X → π(U) = U⋂A where U ⊂ X and π is the retraction : X → A.

Yes I agree with this.

[gentle osprey]

slimvesus:

[dire warren]

Say you take the unit circle in R^n setminus {0}, n >= 2, and smoothly deform it via a smooth homotopy. Now put it back into R^n.. if a subgroup of (R^n, +) contains this deformed circle, is it necessarily all of R^n

[uncut surge]

sheesh

This at least fails for n = 1, where your unit sphere is {1,-1} and thus you could just take the subgroup to be Z

[dire warren]

Lol everyone loves to mention that case 😦

Once and for all I declare n >= 2

[uncut surge]

Yeah it's the only one which is fun

[gentle osprey]

Token:

Token:

[marsh forge]

just take the circular path to the correct angle

and then go up or down

i.e. for any point on the annulus, you can go straight to one of the edges, go around until you get the desired angle, and then go back to the right radius

thats a well defined path and is clearly continuous

[dire warren]

You can also use the ivt if you really want to

[marsh forge]

what

[dire warren]

On your f(t)

[marsh forge]

Oh sure ig

[dire warren]

To show that it must intersect the circle

Pretty overkill but ye

[marsh forge]

is this for a class or for fun

because if you have a really nitpicky professor then maybe you have to explicitly write all this out

but the path is like, very clear

I'll draw a picture

technically when two points are directly opposite you need to just assert you mean clockwise or something

but this is a perfectly good proof of path connectedness

huh????

how is this related to jordan curves

Sorry?

If you have a point in an annulus

and you take the straight path to the origin

you will intersect the circle by IVT or just by explicitly computing when this will happen

it doesn't have anything to do with the jordan curve theorem

Like let $v$ be a vector with $r\leq |v|\leq 1$. Then you can shrink $v$ by taking $tv$ for $t<1$. At some point $|tv|=t|v|$ will be $r$ before it hits $0$

[gentle osprey]

MaxJ:

[marsh forge]

this is just IVT

[coarse kestrel]

Is closure of the intersection the intersection of the closure?

I didn't see the whole argument

But that's the only thing I caught

[coral pivot]

i think its not necessarily true

let me construct counter example hmm

(one inclusion is true btw)

[chrome dew]

[0,1) and (1, 2]

the intersection is empty, so closure of the intersection is empty

[floral gust]

Yeah I realised the equality doesn't hold so I removed the argument

[midnight jewel]

even worse example: Let 𝕀 := ℝ \ ℚ
then cl(𝕀 ∩ ℚ) = ∅ but cl(𝕀)∩cl(ℚ) = ℝ

[coarse kestrel]

Lmao I wasn't asking a question, I was pointing out a problem in someone else's argument

They deleted it before I was able to finish

I forgot who it was tho

[coral pivot]

oh oof

[coarse kestrel]

and also

I'm pretty sure this was an exercise in rudin

[coral pivot]

oh lol makes sense

[coarse kestrel]

Oh for rudin it was union

[coral pivot]

i think for union it is equal right

[coarse kestrel]

Closure of finite union is the finite union of closure

Not infinite case

[coral pivot]

right makes sense

[coarse kestrel]

Yeah

[coral pivot]

because infinite union of closed sets need not be closed

[coarse kestrel]

Exactly

[coral pivot]

so infinite union of closure not closed which is lol

[coarse kestrel]

Is every metric space homeomorphic to a subspace of R^n for some n?

[chrome dew]

don't make me angry

[bitter yoke]

I mean, for one you can have metric spaces on sets that have cardinality larger than R^n so definitely no

[rugged swan]

take a discrete space of cardinal higher than R and the trivial distance

yes

[bitter yoke]

In fact, there are even finite sets that you can't do this for

at least, this is what wikipedia says

[loud scarab]

Hi, I have a question about orientability of surfaces, why do we require the Jacobian of a change of parameters be positive? Doesn't the Jacobian being negative just mean that the normal vector at that point is the just the negative of the original? Which is fine since there exists 2 normal vectors to a tangent plane?

[chrome dew]

it's just a convention that you have to maintain

it's impossible to do on nonorientable surfaces like a mobius strip, once you go around you end up where you started but with opposite orientation, it's inevitable

[loud scarab]

but what does 'going around' mean

[chrome dew]

you're covering it with a family of coordinate neighborhoods

[loud scarab]

why can't i also pick a convention for the mobius strip

[chrome dew]

and at the intersection between two neighborhoods they have to agree by definition you just posted

have you ever seen a mobius strip in real life

maybe think of it this way, orientable surfaces have two sides

[loud scarab]

I saw the ant picture of it

[chrome dew]

nonorientable surfaces don't

[loud scarab]

so the key here is that

[chrome dew]

if you try to paint an orientable surface you could paint one side red and the other side blue if you're imagining the normal vectors outwards and inwards

if you try to paint a mobius strip, you can't paint it with 2 colors because it only has one side

there's no way to distinguish in a global way the difference between normal vectors

maybe try drawing a picture of a mobius strip with a covering on it and try to orient each of them around the entire thing

[loud scarab]

a point on the mobius strip has two neighborhoods that are opposite to each other, thats why, correct?

[chrome dew]

yeah

[loud scarab]

i was trying to apply the same idea to the sphere

the inside of the sphere

is not parametrized?

a hollow sphere of course

[chrome dew]

a surface is 2 dimensional

there's no "inside" points

the only inside it has are the points of the surface itself

[loud scarab]

i mean

[chrome dew]

the "inside" points aren't parametrized because they lie in 3D space which is just an artifact of how you think of the sphere

there is no external space to the sphere really

[loud scarab]

yeah im switching between the mobius strip and the sphere right now in my brain and trying to convince myself

[chrome dew]

it's kind of the opposite problem of how pictures of the klein bottle end up showing a self-intersection

[loud scarab]

i kind of got it but need more time

[chrome dew]

that's not really there either for the same reason that how we immerse it in space is just to try to convey the connectivity

klein bottle is a nice example too, see if you can determine if that's orientable or not

also a torus is another example to work out

[loud scarab]

the torus looks orientable to me

[chrome dew]

yup

easy way to find out if it's nonorientable is to imagine tracing out a path while carrying a normal vector that leads to the same spot with your normal vector flipped

[loud scarab]

alright I finally got it, thank you so much!

[chrome dew]

you're welcome

[dire warren]

Let N be the space of continuous nowhere differentiable functions on the interval, under the subspace topology induced by the sup norm on C[0, 1]. Show that its homotopy groups are identically zero.

(it is path connected so no need to worry about basepoint)

[pine heath]

Topology

[marsh forge]

U stole this from facebook

[uncut surge]

maybe it's their own facebook page

[marsh forge]

If so 22 doxxed himself

[bitter yoke]

It's not his real name nor pfp

[dim meadow]

@dire warren are you James Baxter?

[floral gust]

Can somebody recount how can VA (in a topological group) be considered an open set where A is open and V is closed? I am assuming one will argue by considering VA = ⋃_{v in V} vA and argue vA is open (since a to va is a homeomorphism).
The problem I have here is that I don't need to use closedness of V. So does this hold for any arbitrary subset V or is the above argument wrong?

[frosty sundial]

the argument looks fine to me

but what do you mean "be considered an open set"

something is either open or it's not

it's not a matter of consideration

[marsh forge]

What if it feels open

[coarse kestrel]

Is $\bC$ a universal covering of $\bC\setminus\brc{0}$ by the map $z\mapsto e^z$?

[gentle osprey]

Whoever:

[coarse kestrel]

Where both have the usual metric topology ofc

[coarse kestrel]

Yeah I can see it too, I just want to make sure

[fading vale]

its open in my heart

[exotic root]

Stupid question: let pi(p) = q. The pushforward sheaf pi_*F maps naturally between the stalks by sending an element in f in pi_*F(U) with (f, U)_q = [f]_q being the image in the stalk to the element (f, pi^-1(U))_p or am I missing something?

[loud scarab]

Hello, what's the difference between local diffeomorphism and diffeomorhpsim? When I looked it up, it says that local diffeomorphism that the function restricted to its image is a diffeomorphism, but isn't a diffeomorphism already bijective?

[midnight jewel]

basically the two spaces you’re mapping between may not be diffeomorphic themselves, but the function can still act like a diffeomorphism on small sets. I.e. perhaps f itself isn’t a diffeo, but f restricted to some small open set U is a diffeo onto f(U). if you can find such a U around every point then it’s a local diffeo

obviously every diffeo is a local diffeo as well since you can just choose U to be the whole space

as an example, the map t↦eⁱᵗ between ℝ and S¹ is a local diffeomorphism (pick for U some interval of length less than 2π) but not a diffeomorphism since it’s not bijective

@loud scarab

[loud scarab]

oh okay got it, that example made it clear, thank you

[thin scarab]

so like. if Q is a metric space that has no interior how is it open in itself

nvm i think i understand

[winter rose]

what do you mean by "no interior"?

[dire warren]

For a compact connected n-manifold with/without boundary, what can be said about the least number of connected coordinate charts needed to cover the manifold? Can one find an upper bound that holds uniformly for all such n manifolds for fixed n?

[supple locust]

you cannot have just one coordinate chart cause that will make it homeomorphic to open set $U \subset \mathbb R ^n$ which is not compact. You can have 2 ($S^{n-1} $).

[gentle osprey]

bertwit:

[dire warren]

What about an upper bound?

[supple locust]

i am not sure

are you asked to find an upper bound? i.e. do you know if any upper bound exists?

btw

[dire warren]

Yeah I’m not sure

I accept inf as an answer

Cause that still tells us something

Ofc inf is only an acceptable answer if it’s the lowest upper bound, otherwise inf is trivially an upper bound..

[uncut surge]

related concept: https://en.wikipedia.org/wiki/Lusternik–Schnirelmann_category

also related answer: https://math.stackexchange.com/questions/2097690/minimal-number-of-charts-covering-a-manifold?rq=1

[dire warren]

It seems that requiring the charts to be connected but not contractible hits the sweet spot where it’s not known if an upper bound exists LOL

Well at least not easily found online

[uncut surge]

ah, i see! coordinate charts aren't necessarily contractible

[dire warren]

Ye

[uncut surge]

hm, but i guess there should be something like an inequality between your constant and this LS category

[dire warren]

I was surprised that at the most general they’re not even connected

[uncut surge]

cause nullhomotopic implies connected

[dire warren]

In general not really

Like if u take this LS category

The genus of a surface is a lower bound for the number of nullhomotopic charts

So it can be unbounded even for n = 2

On the other hand a genus n surface can easily be covered with 2 connected charts

Slice it in half

[uncut surge]

Really? That works?

I can't picture that somehow...

[dire warren]

At least for low n I have a picture..

[uncut surge]

For the torus for example, if you sliced that boi, you'd get two cylinders

[dire warren]

Eh

I mean to cut it like

Imagine u take samurai sword

[uncut surge]

oh

[dire warren]

And chop

The whole shit

[uncut surge]

ah yes

that works

[dire warren]

Could prolly demonstrate an explicit homeomorphism

[uncut surge]

cray cray

[dire warren]

With coordinates

But fuck that as maxJ says

[uncut surge]

Alright, so your number can stay at 2 while the LS number goes to infinity

[dire warren]

Ye

[uncut surge]

Does your number <= LS at least?

[dire warren]

... errr

Yes?

Because nullhomotopic charts r connected

So u can always just like

[uncut surge]

Ye okay 😄

[dire warren]

Use those ahah

[uncut surge]

Yeah that was what I was hoping

Okay, but of course that doesn't help you with understanding whether your number might be bounded for all manifolds for a fixed dimension

[dire warren]

Yeah haah cause the sup for LS is in general always inf

[uncut surge]

Alright, that's pretty neat.... so for n = 2, your number is pretty certain to be 2, right?

[dire warren]

Remind me again

[uncut surge]

I guess one might wanna think about surfaces with boundary

[dire warren]

Classification of surfaces

Every closed surface is a n-genus surface or..?

Is it fully classified by the genus?

[uncut surge]

Genus and boundary components

[dire warren]

Right so if no boundary then it’s 2

[uncut surge]

(and maybe orientability? oh man)

[dire warren]

O..

Yeah Möbius strip whoops

yeah idk if it’s in general 2 then ..

Is there a nice classification of non orientwble surfaces without boundary

[uncut surge]

yeah i've got nothing, but i can imagine you can build some weird klein-bottles with higher genus

"A non-orientable surface of genus h can be obtained by gluing h crosscaps to S2. For this, embed D2 in S2 (or Rh−1 from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary S1. The result of attaching a non-orientable handle to S2 or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps."

Unfortunately these are a bit ass to imagine

[dire warren]

Theheck is a crosscap

LOL

Topology is so hard..

[uncut surge]

Like a disk, but the interior of the disk is twisted somehow

[dire warren]

I don’t know how anything gets proved rigorously in topology

[uncut surge]

I wonder if the samurai sword approach still works there...

[dire warren]

this shitvis some shit

LOL

In general, idk

I seem to recall something like

If u (samurai) cut a Möbius strip

You get a connected strip

[uncut surge]

Yeah you're just making it worse

[dire warren]

It was in my childhood book

[uncut surge]

Or wait

Yeah true that was it

I thought there was also a weird way to cut a mobius strip that makes you get two strips entangled or so

[dire warren]

Yeah that’s true

But even then those strips are circular

[uncut surge]

tru

[dire warren]

They won’t be charts

[uncut surge]

Fun question! Might be worth a paper or at least a mathoverflow post

[dire warren]

Yeah I might post a mathoverflow question

It’s certainly beyond me Idk enough topology or geometry or wtv the heck u need

[dire warren]

Oh Larto I have a fun one for you

Let N denote the space of nowhere continuous functions [0, 1] -> [0, 1] under the sup norm topology.
Show that N is path connected.

@uncut surge

[uncut surge]

i swear if this is some baire space shit

it's probably not

that makes me even more mad because you probably need some awful construction 😦

aaaah nooo it doesn't do anything because you can make path-connected spaces disconnected by taking out meagre sets

suffering is at hand

[dire warren]

I did use Baire as a tool at some point..

But it’s not the main ingredient I guess

I only needed that the intersection of two open dense sets is open dense

Also, since it’s path connected, this gives us the go ahead to try computing homotopy and homology groups of this space. Yay! Fun!

[gritty widget]

yoooooo

can anyone help me with this flat earther

[marsh forge]

No one can help a flat earther

[gritty widget]

yes

am i right?

@marsh forge

[marsh forge]

I will not help you with this

[gritty widget]

why

[marsh forge]

Because you should give up

For your own sake

[gritty widget]

I dont care i wanna know if i am right

@marsh forge

[marsh forge]

Dont ping me please

[gritty widget]

then help me

wled le kahb

@MarxJ is a cocycle

@MarxJ is a cocycle

@MarxJ is a cocycle

@MarxJ is a cocycle

@MarxJ is a cocycle

@MarxJ is a cocycle@MarxJ is a cocycle@MarxJ is a cocycle

@MarxJ is a cocycle

@MarxJ is a cocycle

v

@MarxJ is a cocycle@MarxJ is a cocycle

[marsh forge]

Its always an elon fan

[uncut surge]

worst fandom

[ivory dragon]

does idolizing a billionaire megalomaniac count as a "fandom"

[uncut surge]

ah matter of nuance

but i agree with the sentiment

[knotty pasture]

$\partial _{\nu }F^{\mu \nu }=0$

[gentle osprey]

Davide:

[uncut surge]

yes

[gritty widget]

probably stupid question but how is this true?

[gritty widget]

a bijection which is both ways C^p?

hm

oh they're banach spaces

[gentle osprey]

slimvesus:

[gritty widget]

bit only from $\varphi_i(U_i\cap U_j)\to\varphi_j(U_i\cap U_j)$

[gentle osprey]

val:

[gritty widget]

*but

[gentle osprey]

slimvesus:

[gritty widget]

how?

sorry my background here is lacking lmao

[sleek thicket]

you can identify the tangent space of an open subset of V with V

(I'm assuming we're working with manifolds? I saw someone say banach spaces above so I might be out of my element. Sorry if so)

[gritty widget]

ya sorry i should send more context

[sleek thicket]

hmm okay

[gritty widget]

oh nice

well yes

anyway i'll go read the first chapter more carefully

ty

how does that give a map from E_i \to E_j though

yes

[sleek thicket]

Does this book define manifolds as locally a banach space and not locally Euclidean? So you can have like infinite dimensional manifolds?

[gritty widget]

yes

[sleek thicket]

That seems cool

[gritty widget]

m

ok

i think i understand

i will read chpt 1 more closely

ty

yes

where should i go then

[sleek thicket]

what do you know Val?

Like, do you have any background in analysis?

(not trying to be mean)

[gritty widget]

first 6 or so chapters in baby rudin e

ok thanks i'll check it out

lmao