Questions tagged [gn.general-topology]

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

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Points in the Stone Cech compactification are intersection of open sets

Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
4 votes
1 answer
146 views

Given $f$ from the cylinder $C$ to the interval constant on one boundary, is there a $r:C\to C$ constant on a boundary with $f\circ r = f$?

My question might be trivial, but my lack of knowledge of this particular subject has not enabled me to find the answer. What I want to know is the following. Let $I=[0,1]$ and $C=S^1\times I$ be the ...
4 votes
1 answer
264 views

Fiber-bundle : continuity of transition maps and inverse in general

Let $(E,\pi,B)$ be a locally trivial fibration, with fiber a topological space $F$, $\Phi_i$ and $\Phi_j$ two trivializations over $U_i$ and $U_j$. The transition map from $i$ to $j$ is the ...
-4 votes
1 answer
303 views

Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
15 votes
1 answer
748 views

What is this equivalence relation on topological spaces: there are bijective continuous maps in both directions

Consider the following equivalence relation on topological spaces: $X\sim Y$ $:\Longleftrightarrow$ there are bijective continuous maps $\phi:X\to Y$ and $\psi:Y\to X$. Note that there are no ...
3 votes
1 answer
1k views

"Relative compactness of a family of probability measures" and relative compactness & sequential compactness of sets

I'm studying Billingsley's convergence of probability measures, and wondering why the definition of "Relative compactness of a family of probability measures" reasonable. In the discussion ...
9 votes
9 answers
4k views

Help me with this proof: Drop a printed map of the land on the land and there must be some common point.

Hi, I have a minor in math and this is not a homework problem - my prof mentioned it 5 years ago and I could not even begin to tackle it until I took a good intro to linear algebra (after work). ...
11 votes
1 answer
578 views

On the classification of second-countable Stone spaces

Let $X$ be a Stone space (i.e. totally disconnected compact Hausdorff). Then the following are equivalent: $X$ is second countable $X$ is metrizable $X$ has countably many clopen subsets $X$ is an ...
2 votes
1 answer
126 views

The separability of superextensions

The superextension $\lambda X$ of a compact Hausdorff space $X$ is the space of maximal linked systems of closed subsets of $X$, endowed with the Vietoris topology inherited from the double hyperspace ...
10 votes
1 answer
343 views

Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?

Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures. Consider the endomorphism $\hat{\Phi}$ ...
14 votes
2 answers
481 views

Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?

Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder. The map $j:n\...
5 votes
1 answer
255 views

Are Euclidean spaces $\Delta$-generated?

From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$. However, the ...
11 votes
4 answers
2k 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 $\mathbb{N}$?
1 vote
1 answer
149 views

Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?

It seems too good to be possible, but: Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space? Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
0 votes
1 answer
170 views

A continuous injection from the Hilbert cube to the real line?

Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question: Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
11 votes
1 answer
352 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 ...
8 votes
1 answer
257 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 ...
1 vote
2 answers
476 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 votes
1 answer
198 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 ...
6 votes
0 answers
154 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 ...
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 $...
23 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,...
4 votes
1 answer
240 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) ...
32 votes
1 answer
2k views

What happened to the last work Gaunce Lewis was doing when he died?

In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
2 votes
1 answer
266 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 ...
10 votes
1 answer
414 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 ...
4 votes
0 answers
235 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 ...
0 votes
0 answers
77 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 ...
7 votes
3 answers
312 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. ...
6 votes
1 answer
186 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 $W = \{ \text{weak homotopy equivalences} \}$, ...
3 votes
1 answer
119 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 $\{\theta\in\Bbb R^n\mid\|\nabla g(\theta)\|<\eta\}$ ...
1 vote
1 answer
192 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
2 votes
2 answers
235 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 vote
0 answers
64 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) ...
7 votes
5 answers
563 views

Countable chain condition in topology

A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. ...
1 vote
1 answer
145 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 $X$ ...
7 votes
3 answers
884 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 ...
11 votes
1 answer
937 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 ...
4 votes
1 answer
174 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?
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}(...
2 votes
1 answer
313 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 ...
3 votes
1 answer
180 views

Existence of a quasi-open (a.k.a semi-open) map into a Cantor cube

Recall that a topological space is extremally disconnected if the closure of any open set is open. A continuous map is quasi-open if it maps nonempty open sets onto sets with nonempty interior. For ...
10 votes
0 answers
342 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 ...
2 votes
0 answers
163 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
159 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:\...
3 votes
0 answers
129 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 ...
3 votes
1 answer
254 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 ...
2 votes
0 answers
70 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 ...
3 votes
1 answer
206 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 (“...
3 votes
0 answers
90 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 ...

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