Starred posts

Martin Sleziak

Mar 28, 2022 12:04

20

A: What is your favorite proof of Tychonoff's Theorem?

I like the proof from Alexander's subbase lemma. E.g. A proof here. That lemma also gives the compactness criterion in ordered spaces (completeness implies compactness).

Martin Sleziak

Nov 19, 2021 05:19

So basically one way to do this would be to verify that $\mathcal C=\{A \subseteq X : A \cap F \subseteq^{cl} F \forall F \in\mathcal{F}\}$ fulfills the usual axioms for closed sets.

Simple

Nov 6, 2021 23:43

for a given space $X$, define $S_1(X)$ to be the free abelian group with basis all paths $\sigma:I\to X$ and let $S_0(X)$ be the free abelian group with basis $X$. If $x_1,x_0\in X$, show that $x_1-x_0\in im\partial_1$ iff $x_0,x_1$ lie in the same path component of $X$

Martin Sleziak

Oct 19, 2021 13:28

The Wikipedia article Box topology includes Brian M. Scott's answer as a reference: Difference between the behavior of a sequence and a function in product and box topology on same set

Simple

Oct 14, 2021 03:25

Let $\mathbf{R}:S^{1}\to S^{1}$ be rotation by $\alpha$ radians. Prove that $\mathbf{R}\simeq 1_{S}$, where $1_S$ is the identity map of $S^{1}$. Conclude that every continuous map $f:S^{1}\to S^{1}$ is homotopic to a continuous map $g:S^{1}\to S^{1}$ with $g(1)=1$.

Martin Sleziak

Feb 7, 2021 11:08

Martin Sleziak

Feb 5, 2021 06:25

Yes, that's the point I am trying to make. If we work with the subbase, we don't have to deal with each of various special cases (such as ordered set with the smallest element, ordered set with the largest element) separately.

Martin Sleziak

Jan 22, 2021 02:21

Jan 9 at 6:07, by Martin Sleziak

This makes it similar to span, convex hull, subgroup generated by some set, etc. (Linear span is the intersection of all linear subspaces containing the given set. Similarly for convex sets, subgropus and - in this case - topologies.)

Simple

Jan 21, 2021 01:53

Suppose $Z\subset Y$ is $G_{\delta}$ in $Y$. Show that there is a $G_{\delta}$ set $W\subset X$ such that $Z=Y\cap W$.

Martin Sleziak

Jan 20, 2021 04:42

For example, Convergent sequence in co-countable topology iff sequence is eventually constant, Cocountable topology and limits of sequences

Simple

Jan 8, 2021 22:48

Let $F$ be any family of subsets of a space $X$. Show that there is the smallest topology $\tau_F$ on $X$ containing $F$.

Martin Sleziak

Oct 24, 2020 07:49

@Tyrone: That’s a common notion, but I think it a mistaken one: metric spaces are far too nice to be a good place to develop one’s intuition for topological properties. I actually learned topology via a slightly modified Moore method, starting with the definition of a topological space and gradually adding properties that make spaces ‘nicer’, and I still prefer this approach. I recognize, however, that many students are uncomfortable with it and want something familiar as a touchstone. For that purpose I prefer linearly ordered spaces to metric spaces: some familiar examples are readily ... — Brian M. Scott yesterday

Martin Sleziak

Oct 24, 2020 07:49

... available, but there is much less temptation to generalize from them to topological spaces in general than there is with metric spaces, because the fact that they are special is much more apparent. — Brian M. Scott yesterday

Martin Sleziak

Oct 21, 2017 09:29

For instructions how to render MathJax(TeX) in chat see this post on meta.

2

Martin Sleziak

Mar 13, 2020 06:24

A reminder that info on MathJax in chat can be found in this post on meta or the bookmarklet can be obtained directly from robjohn's website.

Math_Is_Fun

Mar 12, 2020 18:39

Let X and Y be connected spaces. Give X X Y the product topology. Then
X X Y is connected. Does the converse of this hold?

user131753

Feb 2, 2020 17:42

0

Q: What are the necessary and sufficient conditions on a function between two topological spaces such that it satisfies the following property?

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be two topological spaces and $f:X\to Y$ is a function such that for all $A,B\subseteq X$, $$f(A)\subseteq \overline{f(B)} \implies A\subseteq \overline{B}$$where $\overline{f(B)}$ denotes the closure of $f(B)$ in $f(X)$ and $\overline{B}$ denotes the closure of ...

YuiTo Cheng

Apr 25, 2019 09:05

Let $G$ be a topological group, $H_1\subset H_2$ be subgroups of $G$ (not necessarily normal). Does the compactness of $G/H_2$ and $H_2/H_1$ imply that of $G/H_1$?

user193319

Mar 4, 2019 12:29

Quote from Hatcher: "If $U$ is connected, these sheets are the connected components of $p^{-1}(U)$ so in this case they are uniquely determined by $U$, but when $U$ is not connected the decomposition of $p^{-1}(U)$ may not be unique...The number of sheets over $U$ is the cardinality of $p^{-1}(x)$ for $x \in U$. As $x$ varies over $X$ this number is locally constant, so it is constant if $X$ is connected."

Arthur Fischer

Dec 5, 2014 11:55

Testing out something here. If I'm correct we'll get postings about new articles in the arXiv's math.GN section. Might be able to save @MartinSleziak the trouble of having to post to keep the room from freezing. We'll see if it works.

2

Martin Sleziak

Oct 2, 2018 06:19

in Modern Abstract Analysis, Sep 27 at 19:12, by Michael Greinecker

Anyone know a general topology book that goes deep into its subject and has a somewhat categorical flavor with lots of diagram chasing?

Poline Sandra

Jun 16, 2018 16:48

For any $a\in \mathbb{R}^*_+$ and $b,c\in\mathbb{R}$ we put $$D_{a,b,c}=\{(x,y)\in\mathbb{R}^2, y>ax+b; y>-ax+c\}$$
we define $\mathcal{B}$ the familly of all the sets $D_{a,b,c}$

let $\theta$ the familly from $\mathbb{R}^2$ which we can represent it by the union of sets from $\mathcal{B}$. prove that $\theta$ is a topology on $\mathbb{R}^2$ with $\mathcal{B}$ is it's basis

Martin Sleziak

Jun 16, 2018 07:14

It is probably so. With this choice of $w^n_j$ we can shown that the sequence $(y_j)$ is cofinally Cauchy, since the restriction to $K_n$ is Cauchy.

Martin Sleziak

Jun 16, 2018 07:10

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

jjepsuomi

May 25, 2018 09:50

I have a simple question. What does "weak compactum" mean? I know that compactum means a complete metric space, but what does "weak compactum" mean?

Martin Sleziak

Feb 6, 2018 16:40

@BAYMAX There certainly is not such homeomorphism. One of these two spaces is compact, the other is not.

Martin Sleziak

Jan 28, 2018 02:09

For instructions how to render MathJax(TeX) in chat see this post on meta.

user131753

Dec 24, 2017 14:02

Does anyone know any criteria of characterizing completely regular spaces expressed only in terms of open sets or closed neighbourhoods?

user193319

Dec 20, 2017 12:56

Okay. From my understanding, given a vector space, the topology with respect to which all semi-norms is continuous is the finest locally convex topology. But what about just all norms, and how does it relate to the finest locally convex topology?

Martin Sleziak

Dec 12, 2017 01:38

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

vincenzoml

Dec 7, 2017 14:06

I'm trying to define a notion of path in Cech closure spaces that specialises to paths in topology and to "graph-like" paths in quasi-discrete closure spaces (that is, those where the closure of any set is the union of the closure of its singletons).
The difficult bit is that, even if I try to use the definition of topological paths via compact Hausdorff spaces, open sets and open neighbourhoods are everywhere and these don't work at all in quasi-discrete closure spaces

Maneesh Narayanan

Nov 11, 2017 07:40

Where will I find the current works in topology? I just completed M.Sc, when I go through some journals, The literature seems strange. I love topology in the graduate level. I studied basics of manifold theory and general topology. I wish to write a statement of purpose. Please help me.

Manolis Lyviakis

Oct 31, 2017 10:47

i wanna see a proof that Q dense in R means there exist those numbers

Manolis Lyviakis

Oct 31, 2017 10:44

but i wanna use only topological tools

Manolis Lyviakis

Oct 31, 2017 10:43

ok i already knew that since Q is dense in R there exist those numbers

Martin Sleziak

Oct 29, 2017 09:35

@user193319 Your proof seems ok to me. If you want to be very pedantic, you can also mention that you proved that $f$ is continuous at $x$. And since $x$ is arbitrary, the conclusion is that $f$ is continuous.

user21820

Oct 26, 2017 14:49

@MartinSleziak Well one could search for the username since a targeted reply must contain it. In any case, that user posted a question on the main site.

Martin Sleziak

Oct 21, 2017 04:50

So if I have product of two sets and one of them is singleton, then projection from $X\times Y$ is bijection.

Martin Sleziak

Oct 21, 2017 04:50

If neither of them is singleton, then it is not a bijection.

user131753

Dec 4, 2016 13:47

Although Brain M. Scott in this question of mine answered it, I was wondering what exactly is wrong with my initial idea of an open injective map being a topological analogue to homorphism.

Martin Sleziak

Apr 16, 2016 16:30

$(X,d)$ is complete if and only if for every sequence $(x_n)$ this implication holds: $$\text{$(x_n)$ is Cauchy with respect to the metric $d$}\implies\text{$(x_n)$ is convergent with respect to the metric $d$}$$

Alessio Ranallo

Apr 3, 2016 09:44

Hi, I have a question about general topology.
Given a topological space X, the space of quasi-components of X is a quotient of X.
Is it true that this space of quasi-components is totally separated?

Martin Sleziak

Feb 8, 2016 08:43

You can also have a peek into Wilard's book on Google Books: books.google.com/books?id=UrsHbOjiR8QC&pg=PA59 (Or you cna try to find it elsewhere on the internet.)

Martin Sleziak

Feb 8, 2016 08:24

But Willard also has a chapter on quotient spaces. (Followed by a set of exercises.) I am not sure whether you would consider Willard's book too advanced, too.

Martin Sleziak

Feb 8, 2016 08:23

@JKnecht Engelking obviously has enough stuff about quotient spaces. But that might be too advanced book.

Martin Sleziak

Feb 8, 2016 07:49

Did you have a look on book recommendations for general topology. Some such questions are listed in List of Generalizations of Common Questions

Martin Sleziak

Feb 8, 2016 07:47

@JKnecht I think that any decent textbook on general topology will include material about quotient spaces.

Arthur Fischer

Oct 18, 2015 14:07

@MartinSleziak This isn't quite accurate. Austin Mohr originally developed something he called "SpaceBook". About the same time James Dabbs had developed a different topological space tool (the original name of which I forget). That was off-line for quite a period of time during which it developed into π-Base. Austin Mohr seems to have abandoned his Spacebook in favor of π-Base.

Arthur Fischer

Oct 18, 2015 14:07

Here's a write up from SciAm online.

Martin Sleziak

Jun 30, 2015 11:36

Mary E. Rudin: "Set theory and General Topology"

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General topology

General topology