Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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67
votes
9answers
19k views

solving f(f(x))=g(x)

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
10
votes
5answers
3k 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 ...
12
votes
6answers
2k 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 ...
30
votes
4answers
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 $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ ...
49
votes
2answers
4k 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 ...
46
votes
8answers
5k 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 ...
22
votes
3answers
5k 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 ...
27
votes
8answers
3k 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 ...
18
votes
5answers
6k views

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

Here I mean the version with all but finitely many components zero.
27
votes
4answers
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 ...
19
votes
6answers
2k 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. ...
14
votes
6answers
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 ...
11
votes
1answer
318 views

Strongly rigid Hausdorff spaces

A space $(X,\tau)$ is called rigid if $\textrm{Aut}(X)=\{\textrm{id}_X\}$. We say $(X,\tau)$ is strongly rigid if for every continuous map $f:X\to X$ we have that $f = \textrm{id}_X$ or $f$ is ...
3
votes
1answer
489 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\{ ...
15
votes
3answers
435 views

Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?

This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...
6
votes
4answers
576 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 $\mathbb{N}$?
3
votes
1answer
132 views

Properties of the interval topology of the lattice of functions

Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: ...
44
votes
9answers
11k 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 ...
68
votes
4answers
11k 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 ...
44
votes
2answers
4k 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 ...
50
votes
11answers
5k 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" ...
28
votes
8answers
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 ...
23
votes
17answers
7k 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: ...
42
votes
4answers
3k views

Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining. Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...
38
votes
8answers
4k 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 $g: Y \mapsto X$?
23
votes
4answers
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 ...
35
votes
2answers
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 ...
23
votes
6answers
3k views

Why the triangle inequality?

[Maybe this is asking to be closed; but I thought I'd risk it.] A metric satisfies the axioms: $d(x,y)=0$ if and only if $x=y$. $d(x,y) = d(y,x)$. $d(x,y) \leq d(x,z) + d(z,y)$. Similarly (and ...
23
votes
4answers
1k views

is f a polynomial provided that it is “partially” smooth?

Let $f$ be a $C^\infty$ function on $(c,d)$ ,and let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$. Suppose for each $n\in ...
23
votes
4answers
3k views

Topological Characterisation of the real line.

What is a purely topological characterisation of the real line( standard topology)?
12
votes
1answer
960 views

Questions about spectra of rings of continuous functions

I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
19
votes
3answers
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. ...
11
votes
7answers
2k 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 ...
9
votes
4answers
1k 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
3answers
2k views

Topological dimension versus cohomological dimension

This should be really well known but I don't seem to find a statement about it nor a question in MO answering this. Consider a Compact Hausdorff topological space $X$. The cohomological dimension of ...
7
votes
4answers
798 views

nonhausdorff dimension

if $X$ is a topological space, a first step in making $X$ hausdorff is taking the quotient $H(X)=X/\sim$, where $\sim$ is the equivalence relation generated by: if $x,y$ cannot be seperated by ...
14
votes
1answer
1k views

complement of a totally disconnected closed set in the plane

While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
6
votes
2answers
213 views

Coarsest admissible topology on $\text{Cont}(X,Y)$

Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation ...
5
votes
1answer
730 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 ...
4
votes
2answers
495 views

von neumann algebras and measurable spaces

I've read some pages on links between von neumann (VN) algebras and measurable spaces (Spectra of $C^*$ algebras and Non-commutative geometry from von Neumann algebras?), but I can't get the ...
14
votes
1answer
344 views

Nonperiodic points of homeomorphisms of a ball

Suppose $B$ is a $d$-dimensional ball (for some $d \geq 1$) and $T$ is a homeomorphism from $B$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that ...
8
votes
3answers
749 views

Minimal Hausdorff

A Hausdorff space $(X,\tau)$ is said to be minimal Hausdorff if for each topology $\tau' \subseteq \tau$ with $\tau' \neq \tau$ the space $(X,\tau')$ is not Hausdorff. Every compact Hausdorff space ...
6
votes
2answers
481 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
0answers
289 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
3answers
704 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
4answers
646 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 ...
16
votes
2answers
1k views

The letters of the word “ART”

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is ...
4
votes
2answers
212 views

Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...
3
votes
1answer
285 views

Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation ...
2
votes
1answer
123 views

Product of posets with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq x\}$ and ...