red whisker

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$Mathematics$ Discussion on question by Brien Navar…

Imported from a comment discussion on math.stackexchange.com/q...

red whisker

Dec 1, 2023 16:26

Sure, ping me here

red whisker

Dec 1, 2023 16:26

The $n$th derivative of $f$ is the $(n-1)$st derivative of $f'$. If needed, I'll write down a complete answer.

$Mathematics$ Mathematics

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red whisker

Jan 3, 2023 21:20

The basic box neighbourhoods form a basis, so any local basis may be assumed to consist of box neighbourhoods. Now, assuming a countable local basis, do a Cantor diagonalization kind of argument where you create a new basic neighbourhood whose nth interval is half the size of the nth interval in the nth neighbourhood of the local basis.

stackex33

Dec 15, 2020 15:20

I see, interesting. Thanks for your help!

stackex33

Dec 15, 2020 15:15

true

stackex33

Dec 15, 2020 15:14

@MikeMiller of course!

stackex33

Dec 15, 2020 15:10

@MikeMiller one homeomorphism is given by passing z\mapsto z^4 to the quotient, and there's no submersion because the x and y axes have the same image (so the derivative is not injective), that right?

stackex33

Dec 15, 2020 14:37

@MikeMiller @MikeMiller I see, thanks. I'll think about the problem (thanks for the problem as well)

stackex33

Dec 15, 2020 14:32

@MikeMiller what about a manifold with boundary? yeah, well I'm not completely familiar with this

stackex33

Dec 15, 2020 14:30

@MikeMiller Oh, it is problematic?, it looked like a semicircle, so I didn't bother about it

stackex33

Dec 15, 2020 14:29

@MikeMiller I see, I couldn't think of an example where having pinch points was actually problematic

stackex33

Dec 15, 2020 14:26

As for the problems when it's not free, I'm guessing it messes up the Hausdorff condition, an extreme example I can think of is non zero reals acting by multiplication on reals. But in general, I suppose one has certain "pinch points" (I'm thinking of folding the unit disc by conjugation), that may be problematic. I don't know, I'll have to study it

stackex33

Dec 15, 2020 14:24

@MikeMiller well, you're close. I was actually looking at the covering X over X/G and that led me to this. In wiki, they mention free action, and I had heard of it as "fixed point free", so I just wanted to confirm that it meant what I thought it meant.

stackex33

Dec 15, 2020 13:25

@Thorgott sure, just that I had manifolds on my mind

stackex33

Dec 15, 2020 13:23

@BalarkaSen ah, I see, thanks

stackex33

Dec 15, 2020 13:22

What does it mean for an action to be free in the context of groups acting on manifolds? That it is fixed point free?

stackex33

Dec 2, 2020 12:13

Q: Universal cover of the Hawaiian earring

Let $H$ be the Hawaiian earring, i.e. it is the union of circles $S_n$ with centre $(1/n, 0)$ and radius $1/n$. By a universal cover I mean a covering $p\colon X\to H$ (with $X$ connected) such that for any other covering $p'\colon X'\to H$, there is a map $f\colon X\to X'$ such that $p'\circ f =...

stackex33

Dec 2, 2020 12:13

Hello everyone, I have a question math.stackexchange.com/questions/3930308/…

stackex33

Nov 12, 2020 14:34

A: Jump of an infinite step function

Instead of having a constant on $[0, 1/n]$, take a line from $(1/n, 1/n)$ to $(-1, 0)$ and then for $x<-1$, keep all $a_n=0$ (do the same thing for the other family of functions). This way the $a_n$ form a family of continuous functions. Now, fix some integer $n\geq 2$. Since $a_n$ is continuous ...

stackex33

Nov 12, 2020 14:32

Hi everyone, could someone verify my answer here. I posted this on the constructive feedback room but haven't (yet) recieved any feedback (constructive or otherwise). Thanks

$Mathematics$ Constructive Feedback

Feedback and advice to help users improve their questions and/...

stackex33

Nov 12, 2020 04:16

Could someone verify my answer here? The OP seems to have disappeared after halloween. Thanks