All Questions
5,184 questions
11
votes
1
answer
428
views
Is the Mandelbrot set Suslinian?
The Mandelbrot set is known to be (path-)connected and compact. A non-degenerate space with these properties is called a continuum.
A continuum $X$ is Suslinian if every collection of non-degenerate ...
- 6,548
8
votes
1
answer
278
views
What is known about these "explicitly represented" spaces?
Apologies if this is too low-level. A related question that I asked on the Math Stack Exchange got no answers after a year, so I thought it might be better to ask this one here.
The standard approach ...
- 4,727
1
vote
2
answers
484
views
Is there good evidence that topological spaces are the correct way to study the general theory of continuity? [closed]
My reason for asking is that the theory of metric spaces is so clean and so many significant theorems can be proved for an arbitrary metric space (which makes it plausible to me that metric spaces are ...
- 4,713
4
votes
1
answer
223
views
Is every compact, sober, second-countable space the image of $2^\omega$?
As a bonus, is every compact, $T_0$, second-countable space the image of $2^\omega \times \omega$?
As a further bonus, can we strengthen "image" to "quotient"?
My motivation for ...
- 4,727
6
votes
0
answers
169
views
Topological spaces for which $w(X)\leq |X|$ holds
Let $w(X) = \inf\{|\mathcal{B}| : \mathcal{B} \text{ is a base for }X\}$ be the weight of topological space $X$.
For metric spaces and locally compact spaces we have inequality $w(X)\leq |X|$.
This ...
- 1,211
17
votes
3
answers
2k
views
Is symmetric power of a manifold a manifold?
A Hausdorff, second-countable space $M$ is called a topological manifold if $M$ is locally Euclidean. Let $SP^n(M): = \left(M \times M \times \cdots \times M \right)/ \Sigma_m$, where product is done $...
6
votes
1
answer
173
views
References on semigroup actions
I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994).
I would like to ask for references on semigroup actions on ...
- 4,459
24
votes
1
answer
1k
views
What topological principle is at work here?
[I'm cross-posting this from MSE. I initially asked there 10 days ago, and the question was well-received, but left unanswered.]
My question is inspired by a problem I discovered in Putnam and Beyond,...
- 55.9k
4
votes
1
answer
252
views
Does every (Abelian) Polish group have a nontrivial locally compact subgroup?
The question is pretty much in the title, suppose that $G$ is an (Abelian) nontrivial Polish group, must $G$ have a nontrivial locally compact (in the induced topology, hence necessarily closed) ...
- 655
2
votes
1
answer
300
views
If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
- 45k
11
votes
1
answer
444
views
Topological spaces admitting CAT(1) metrics
Suppose that $X$ is a locally contractible completely metrizable topological space. Is it true that $X$ can be metrized as a (complete) CAT(1) metric space?
The only result in this direction I know is ...
- 45k
4
votes
0
answers
249
views
Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation
The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
- 2,858
0
votes
0
answers
117
views
Example of a metrizable space that is not an ANR
I have been looking for an example of a metrizable space that is not an absolute neighborhood retract (ANR).
Recall that a metrizable space $X$ is called an ANR if there exists an open set $U$ in a ...
- 506
7
votes
3
answers
356
views
Hausdorff quasi-Polish spaces
A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article: de Brecht, Matthew, Quasi-Polish spaces, Ann. ...
- 3,612
6
votes
1
answer
231
views
Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
- 30.3k
3
votes
1
answer
132
views
Is it possible to determine whether the critical values are nowhere dense in the case of a bounded set of stationary points?
Let $g:\Bbb R^{d}\rightarrow \Bbb R$ be a non-negative, continuously differentiable function satisfying the following two conditions:
The set $\{\theta\in\Bbb R^n\mid\|\nabla g(\theta)\|<\eta\}$ ...
- 28k
2
votes
2
answers
274
views
Is a simple closed curve always a free boundary arc?
Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points?
For a simple closed curve $\...
- 1,375
1
vote
0
answers
85
views
Increasing coverings of rigid analytic varieties
Let $K/\mathbb{Q}_p$ be complete and let $X/K$ be a rigid analytic variety. When does $X$ admit an "increasing" admissible covering by quasi-compact admissible (in the strong G-topology) ...
- 36
1
vote
1
answer
153
views
For topological torus action, there is a subcircle whose fixed point is the same as the torus
Let $T=\mathbb{S}^{1}\times \mathbb{S}^{1}\times \cdots \times \mathbb{S}^{1}
$ ($n$ times) be an $n$-dimensional torus acting on any topological space $X$.
The group $G$ is said to act on a space $X$ ...
- 12.9k
7
votes
3
answers
911
views
A fibrant-objects structure on Top
(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
- 30.3k
11
votes
1
answer
997
views
How many model category structures are there on Top?
I recently started learning a little model category theory and in particular I found this nice exercise. I only know a little topology, but this prompted me to wonder how many model category ...
- 30.3k
5
votes
1
answer
191
views
Monoidal topology and coarse spaces
Is there a description of (quasi-)coarse spaces that is analogous to the description of (quasi-)uniform spaces as lax algebras?
- 13.6k
20
votes
3
answers
2k
views
Duality between topology and bornology
I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:
Let $X$ be a set and let $\mathcal{P}(...
- 3,816
2
votes
1
answer
345
views
Function series of normal lower semi-continuous functions
For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is
defined as follows:
$
f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in
U\right\} :U\in ...
10
votes
0
answers
354
views
Cellular-Lindelöf: a common generalization of the Lindelöf property and the CCC
All spaces are assumed to be Hausdorff. Recall that a cellular family in the space $X$ is a family of pairwise disjoint non-empty open subspaces of $X$. The cellularity of $X$ ($c(X)$) is defined as ...
- 12.9k
2
votes
0
answers
164
views
Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
5
votes
0
answers
171
views
Length metrics on covering spaces
This is a question (Exercise 3.30(2)) in the book `Metric spaces of non-positive curvature' written by Bridson and Haefliger.
In the book, there is the following proposition (Proposition 3.28)
Let $p:\...
- 45k
4
votes
0
answers
156
views
Known relations between mutual information and covering number?
This is a question about statistical learning theory. Consider a hypothesis class $\mathcal{F}$, parameterized by real vectors $w \in \mathbb{R}^p$. Suppose I have a data distribution $D \sim \mu$ and ...
- 141
3
votes
0
answers
75
views
Are the automorphisms of the power semigroup of a cancellative semigroup cardinality-preserving?
Let $S$ be a semigroup (written multiplicatively) and $f$ be an automorphism of the power semigroup $\mathcal P(S)$ of $S$, that is, a bijective function on the family of all non-empty subsets of $S$ ...
- 10.5k
3
votes
1
answer
268
views
Is the Fortissimo space on discrete $\omega_1$ radial?
Let $\omega_1$ have the discrete topology. Its Fortissimo space is $X=\omega_1\cup\{\infty\}$ where neighborhoods of $\infty$ are co-countable.
A space is radial provided for every subset $A$ and ...
- 1,382
2
votes
0
answers
105
views
What is known about sublocales defined by regular nuclei?
(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.)
I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements ...
- 32.5k
3
votes
1
answer
228
views
Computing the Heyting operation on the frame of nuclei
(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound (“...
- 32.5k
3
votes
0
answers
101
views
Constructively valid reference for the soberness of discrete spaces and points of a locale coproduct
I am looking for constructively valid references for the following two related facts:
discrete topological spaces are sober,
the points of a locale coproduct are the disjoint union of the points of ...
- 32.5k
7
votes
1
answer
185
views
Existence of Borel uniformization for coanalytic set with non-$K_\sigma$ sections
Suppose that $X$ is a Polish (or standard Borel) space and $\omega^\omega$ is the Baire space of all natural number sequences. My question is: If $A\subseteq X\times \omega^\omega$ is a coanalytic set ...
- 9,902
0
votes
0
answers
98
views
Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?
(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...
- 2,830
6
votes
3
answers
2k
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 ...
- 45k
4
votes
1
answer
110
views
Separation of convexity on uniquely geodesic space
A metric $d: X \times X \to [0,\infty)$
is said to be intrinsic provided that the distance between any two points is the infimum of the lengths of
paths joining the points. A space is an inner metric ...
- 45k
2
votes
1
answer
213
views
Parametrization of topological algebraic objects
There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...
CommunityBot
- 1
2
votes
1
answer
108
views
Mandelbrot boundary and component of $\infty$
Let $M$ be the Mandelbrot set, and $\partial M$ its boundary. So $\partial M$ is the set of those points $z\in M$ such that every neighborhood of $z$ contains a point of $\mathbb R^2\setminus M$.
Let $...
- 91.8k
1
vote
0
answers
80
views
Construct manifold given simplical complex
It's known that, in general, given a simplical complex, answering if it's homeomorphic to a manifold is undecidable. However, given a positive answer to the problem, is there an algorithm to construct ...
- 141
4
votes
1
answer
179
views
A "simple" space with closed retracts but non-unique sequential limits
This question asking how KC ("Kompacts are Closed") and RC ("Retracts are Closed") are distinct has some good discussion, including a now-published example by Banakh and Stelmakh ...
- 476
1
vote
0
answers
83
views
Approximating evalutation maps at open sets over invariant measures
Let $G$ be a group acting by homeomorphisms on a compact metrizable space, say $X$; let's denote by $\alpha:G\to\mathrm{Homeo}(X)$ the action, $g\mapsto\alpha_g$, and consider the weak-$^*$ compact ...
- 93
3
votes
0
answers
103
views
An isomorphism problem for semigroups of ideals
An ideal of a semigroup $S$ (written multiplicatively) is a set $I \subseteq S$ such that $IS$ and $SI$ are both contained in $I$ (here, $XY$ means, for all $X, Y \subseteq S$, the setwise product of $...
- 10.5k
1
vote
0
answers
62
views
Continuous maps between compact space and cubes
Let $X$ be a compact metrizable space. Let $f$ be a continuous map from $X$ to the cube $[0,1]^m$. I would like to know under which condition of a continuous map $g: X\to [0,1]^n$ there exists a ...
- 675
3
votes
1
answer
459
views
When is the Freudenthal compactification an ANR?
Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is:
What are ...
- 1,446
4
votes
1
answer
178
views
Compact-open Topology for Partial Maps?
I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow.
Compact open topology is one of the most common ways of ...
3
votes
0
answers
239
views
Metrizing pointwise convergence of *sequences* of functionals in a dual space
This question was asked by myself on the math stackexchange a few days ago. I thought I'd repeat it here:
Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of ...
3
votes
1
answer
203
views
Simple closed curves in a simply connected domain
Let $U$ be a bounded simply connected domain in the plane. Let $K$ be the boundary (or frontier) of $U$. For every $\varepsilon>0$ is there a simple closed curve $S\subset U$ such that the ...
- 60.5k
2
votes
0
answers
136
views
Progess on conjectures of Palis
I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...
- 247
2
votes
2
answers
356
views
Is a Hausdorff separable topological space that is uniform and complete necessarily a Polish space?
Is an Hausdorff separable topological space that is uniform and complete necessarily a Polish space ?
- 1,382