**8**

votes

**6**answers

1k views

### Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here.
Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$.
Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...

**44**

votes

**2**answers

3k views

### Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...

**20**

votes

**3**answers

4k views

### When is $L^2(X)$ separable?

I have never studied any measure theory, so apologise in advance, if my question is easy:
Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable?
In reality, I am interested in ...

**16**

votes

**6**answers

5k views

### How do you show that $S^{\infty}$ is contractible?

Here I mean the version with all but finitely many components zero.

**14**

votes

**6**answers

2k views

### Topological characterization of the closed interval $[0,1]$

This question is related to question 92206 "What properties make $[0, 1]$ a good candidate for defining fundamental groups?" but is not exactly equivalent in my opinion. It is even suggested in one ...

**2**

votes

**1**answer

343 views

### Different Metrics for Baire Space and their induced Topologies

The Baire-Space is the set of all infinite sequences of integers, i.e.
$$
\mathcal N = \omega^{\omega}.
$$
On this space usually the following metric is given
$$
d(\alpha, \beta) = \left\{ ...

**5**

votes

**4**answers

474 views

### Non-trivial convergent sequence in Stone-Čech compactification of $\mathbb{N}$

Why are there only trivial convergent sequences in the Stone-Čech compactification of $\mathbb{N}$?

**48**

votes

**11**answers

4k views

### How should one think about non-Hausdorff topologies?

In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" ...

**27**

votes

**8**answers

3k views

### What is a metric space?

According to categorical lore, objects in a category are just a way of separating morphisms. The objects themselves are considered slightly disparagingly. In particular, if I can't distinguish ...

**43**

votes

**8**answers

4k views

### What is a continuous path?

I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're already getting ...

**32**

votes

**2**answers

1k views

### Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...

**26**

votes

**8**answers

2k views

### When are there enough projective sheaves on a space X?

This question is being asked on behalf of a colleague of mine.
Let X be a topological space. It is well known that the abelian category of sheaves on X has enough injectives: that is, every sheaf ...

**23**

votes

**4**answers

2k views

### Topological Characterisation of the real line.

What is a purely topological characterisation of the real line( standard topology)?

**25**

votes

**4**answers

2k views

### Reference for the Gelfand-Neumark theorem for commutative von Neumann algebras

The Gelfand-Neumark theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann ...

**11**

votes

**7**answers

1k views

### Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...

**8**

votes

**5**answers

2k views

### totally disconnected and zero-dimensional spaces

When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: covering dimension, small ...

**25**

votes

**4**answers

2k views

### Are the rationals homeomorphic to any power of the rationals?

I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ ...

**5**

votes

**1**answer

659 views

### A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite ...

**6**

votes

**2**answers

469 views

### Homotopy problem for infinite dimensional topological space II

This post here is a specification of this post.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying :
$X_{n}$ have topological dimension $n$.
$X_{n+1}$ is ...

**6**

votes

**0**answers

231 views

### Does local strict contractibility imply ANR?

Say that a space (= compact metrizable space) $X$ is locally strictly contractible if, for every $p\in X$ and neighborhood $U$ of $p$, there is a neighborhood $V$ of $p$ which can be contracted to $p$ ...

**4**

votes

**3**answers

525 views

### Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...

**3**

votes

**4**answers

578 views

### On locally convex (and compactly generated) topological vector spaces

Part 1:
How big is the category $TVS_{loc.conv.}$ of locally convex topological vector spaces (and continuous maps)?
In other words (and less cheekily), is there a free locally convex TVS having any ...

**1**

vote

**1**answer

342 views

### The space $\psi$

Is the space $\psi$ (described in problem 5I of L. Gillman and M. Jerison, Rings of continuous functions, Springer Verlag, 1976) a F-Z-space (i.e, space with $cl(X-Z(f))$ is a zero set for every $f$ ...

**-11**

votes

**1**answer

1k views

### Union of uniformly connected sets

I will call a set uniformly connected regarding some uniform space when it is connected regarding every entourage of this uniform space (entourages are considered as digraphs and it is taken strong ...

**44**

votes

**9**answers

9k views

### Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...

**58**

votes

**5**answers

3k views

### Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...

**61**

votes

**4**answers

8k views

### Does the inverse function theorem hold for everywhere differentiable maps?

(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.)
Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...

**38**

votes

**2**answers

3k views

### Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...

**21**

votes

**17**answers

6k views

### Applications of Brouwer's fixed point theorem

I'm presenting Brouwer's fixed point theorem to an audience that knows some point-set topology. Does anyone have any zippy / enlightening / cool applications or consequences of it? So far, I have:
...

**33**

votes

**7**answers

3k views

### Non-homeomorphic spaces that have continuous bijections between them

What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \mapsto Y$ and $g: Y \mapsto X$?

**21**

votes

**4**answers

3k views

### Why are the integers with the cofinite topology not path-connected?

An apparently elementary question that bugs me for quite some time:
(1) Why are the integers with the cofinite topology not path-connected?
Recall that the open sets in the cofinite topology on ...

**26**

votes

**6**answers

2k views

### When factors may be cancelled in homeomorphic products?

It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, ...

**22**

votes

**3**answers

3k views

### Is “compact implies sequentially compact” consistent with ZF?

Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...

**14**

votes

**8**answers

2k views

### References for homotopy colimit

(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...

**32**

votes

**1**answer

664 views

### Homeomorphisms and disjoint unions

Let $X$ and $Y$ be compact subsets of $\mathbb{R}^n$. Assume that $X \sqcup X \cong Y \sqcup Y$ (here $X \sqcup X$ is the disjoint union of two copies of $X$, considered as a topological space, and ...

**11**

votes

**4**answers

7k views

### On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + …

It is well-known that
A: The series of the reciprocals of the primes diverges
My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers.
Property A ...

**26**

votes

**3**answers

1k views

### Can a connected planar compactum minus a point be totally disconnected?

What the title said. In a slightly more leisurely fashion:-
Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x}$ be ...

**12**

votes

**11**answers

1k views

### Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...

**30**

votes

**3**answers

2k views

### Independent evidence for the classification of topological 4-manifolds?

Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? ...

**22**

votes

**3**answers

461 views

### What sets of self-maps are the continuous self-maps under some topology?

An open question on MSE, http://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a ...

**20**

votes

**8**answers

1k views

### Self-containing structures

(This question is partly inspired by What is inter-universal geometry?.)
I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description ...

**16**

votes

**5**answers

1k views

### A topological concept dual to compactness

We say that a subset A in a topological space X is anti-compact if every covering of A by closed sets has a finite subcover. Clearly if X is Hausdorff then all anti-compact subsets of X are finite. ...

**15**

votes

**1**answer

487 views

### Closed totally disconnected subspaces

It is a remarkable property of uncountable compact metric spaces that each of them contains a homeomorphic copy of the Cantor set. In general, one cannot expect containment of Cantor cubes (in ...

**15**

votes

**1**answer

984 views

### connectivity of the group of orientation-preserving homeomorphisms of the sphere

In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...

**12**

votes

**4**answers

1k views

### What is the “right” universal property of the completion of a metric space?

I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes:
Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...

**17**

votes

**3**answers

2k views

### The deep significance of the question of the Mandelbrot set's local connectedness?

I am given to understand that the celebrated open problem (MLC) of the Mandelbrot set's local connectness has broader and deeper significance deeper than some mere curiosity of point-set topology.
...

**14**

votes

**5**answers

1k views

### Which topological spaces are (topological) groups?

General literature does not seem to offer a characterisation of topological groups among all topological spaces. Of course, being completely regular (uniform) is necessary, but separation properties, ...

**9**

votes

**4**answers

943 views

### Elements of infinite order in a profinite group

Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general?
A start for (A): we can ask the same question ...

**7**

votes

**2**answers

398 views

### Potential connected non-Lie subgroup

This painful question is inspired by the question
"non-Lie subgroups" . Let R denote the real numbers. Let f be an discontinuous additive map from R --> R. Is it possible that the graph of f, inside ...

**5**

votes

**3**answers

1k views

### Where can I find a proof of the de Rham-Weil theorem?

Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?