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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Extremally disconnected sets as building blocks for compact Hausdorff spaces

Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
Peter Kropholler's user avatar
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0 answers
99 views

Prove a generalization of hairy-ball theorem

I found an interesting question below: Prove that the 6-sphere admits no continuous field of tangent 3-planes.
threeautumn's user avatar
0 votes
1 answer
60 views

Real exponentiation in the quotients of rings of continuous functions by prime ideals

Consider the ring $C = C(X) = C(X; \mathbb{R})$ of continuous functions $f:X\to \mathbb{R}$ where $X$ is a Tychonoff space. This is naturally a lattice ordered ring by setting $f\geq 0$ iff $f(x)\geq ...
Jakobian's user avatar
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7 votes
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76 views

Is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?

In short, the question is in the title: is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis? A bit of context: Given a topological space $X$, a family $\...
Cla's user avatar
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-4 votes
0 answers
81 views

What is the Measure of the Permutations of the Real Numbers? [closed]

Since the permutations of the real numbers would form a set of cardinality $\aleph_2$, do we just say it has infinite measure? It would seem to be the case for a Lebesgue measure. Is there a standard ...
Carl Gueck's user avatar
2 votes
0 answers
54 views

Union of two open, open-unicoherent sets whose intersection is connected

I stumbled upon the following proposition, and haven't found an error in my proof yet. By "open-unicoherence" I mean unicoherence with closed sets replaced with open sets in the definition. ...
Calvin Wooyoung Chin's user avatar
-1 votes
0 answers
57 views

Convergence of convex compact bodies implies Hausdorff convergence

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that: (simple convergence) for every $x \in \mathbb{...
Anthony's user avatar
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5 votes
1 answer
208 views

In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?

Let $G$ be a (Hausdorff) topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. ...
Linus's user avatar
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1 vote
1 answer
69 views

Simple convergence of convex compact set implies Hausdorff convergence

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
Anthony's user avatar
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1 vote
0 answers
61 views

Uniform approximation over compacts using weighted function spaces

I'm interested in approximations over the so-called weighted function spaces. Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
Gaspar's user avatar
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7 votes
1 answer
330 views

Proving the inequality involving Hausdorff distance and Wasserstein infinity distance

Prove the inequality $$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$ where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
Luna Belle's user avatar
5 votes
1 answer
241 views

Quotients in categories of metric spaces

There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
Jochen Wengenroth's user avatar
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0 answers
137 views

Connectedness of deleted symmetric product

Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
Peluso's user avatar
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$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]

If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$ \left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
Ola Sande's user avatar
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8 votes
1 answer
338 views

The cardinality of projections of subsets of the Hilbert cube by inner products

I have three related questions. Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
Boaz Tsaban's user avatar
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6 votes
1 answer
133 views

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
Alexander Osipov's user avatar
2 votes
0 answers
24 views

Continuity of Kernel Mean Embeddings

Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
Gaspar's user avatar
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0 answers
58 views

"Star" of a CW-complex

Suppose we have a CW-complex $X$ with a 0-cell $e^0$. Is the union of all the cells (of higher dimensions) for which $e^0$ is a boundary point open in $X$? I don't know if it has a name, but a similar ...
brattok's user avatar
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0 votes
1 answer
46 views

Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K?

Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components. Let $U$ be a connected component of $S \setminus K$ and ...
Fernando Oliveira's user avatar
2 votes
1 answer
157 views

Factorization systems for vector bundles

Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
Nash's user avatar
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1 vote
1 answer
106 views

Variants of Dirichlet-type function as a pointwise limit of continuous functions

Problem Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both ...
hmeng's user avatar
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3 votes
1 answer
170 views

Is there a metric separable space with the following properties...?

Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$. Is there a metric separable space $X$ with the following properties: $|X|\geq\...
Alexander Osipov's user avatar
0 votes
0 answers
81 views

Finding an example if it exists, for a non-contractible and contractible space with special requirement on quotients of their union?

Let $A$ and $B$ be subsets of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, such that $A$ is non-contractible, $B$ is contractible and $B$ is not an one-point set. I would like to find example(s) ...
Himanshu Yadav's user avatar
0 votes
0 answers
49 views

Approximating open subset of profinite group by union of cosets of ideal

I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset $U_P \subseteq \hat{\mathcal{O}}_P$ can be ...
jb1403's user avatar
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-4 votes
0 answers
417 views

N dimensional, not-locally Euclidean, non-Hausdorff topological space

Take a topological space $(M, \tau) $ where $\tau$ is the collection of open sets of $M$. Suppose: the Lebesgue covering dimension of this space is $N \geq 1$ Non-Hausdorff Not locally Euclidean The ...
Bastam Tajik's user avatar
3 votes
0 answers
86 views

Finitely generated Banach lattice $C(K)$ and partitions of unity

Let $E$ be a Banach lattice. A Banach sublattice $L$ of $E$ is called finitely generated if there exists a finite subset $F \subseteq E$ such that $$L = \bigcap \{ \hat{L} \mid F \subseteq \hat L, \, \...
Julian Hölz's user avatar
5 votes
0 answers
79 views

When a compact subset of a TVS can be continuously projected on a closed linear subspace?

Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact. (Q): When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
Pietro Majer's user avatar
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0 votes
0 answers
35 views

Estimate the gradient (with respect to local coordinates) of a partition of unity on a manifold

Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume ...
Anar C's user avatar
  • 21
0 votes
0 answers
56 views

Computing the eta invariant of a rather contrived operator on the circle

For physical reasons, I am interested in computing the eta invariant of the following Hermitian operator acting on complex valued functions on the circle with circumference 1. I define the operator ...
Blind Miner's user avatar
1 vote
2 answers
118 views

Axiomatic definition of Katětov closure operator

In the book "Categorical Structure of Closure Operators with Applications to Topology" by Dikranjan and Tholen a Katětov closure operator is defined in terms of filter covergence: $k_X(M):=\{...
Jaŭhien Piatlicki's user avatar
11 votes
1 answer
276 views

Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?

In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$. The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
Pietro Majer's user avatar
  • 56.5k
1 vote
1 answer
123 views

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 ...
Serge the Toaster's user avatar
3 votes
1 answer
565 views

What is the Lebesgue covering dimension of this topological space?

Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal. Take the induced topology defined by the Lorentzian metric called Alexandrov topology. This topology matches ...
Bastam Tajik's user avatar
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 ...
M. Winter's user avatar
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-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 ...
Bastam Tajik's user avatar
1 vote
2 answers
484 views

Show convergence result

Consider the following sets: $$ A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\}, $$ and $$ A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)...
Star's user avatar
  • 64
4 votes
0 answers
275 views

Which countable sets don't drastically change the definable topologies on $\mathbb{R}$?

For $\mathcal{M}$ an expansion of $\mathcal{R}=(\mathbb{R};+,\times)$ and $A\subseteq\mathbb{R}$, let $\tau^\mathcal{M}_A$ be the topology on $\mathbb{R}$ generated by the sets definable in $\mathcal{...
Noah Schweber's user avatar
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 ...
Tim Campion's user avatar
  • 60.6k
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 ...
Taras Banakh's user avatar
  • 40.8k
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 $[0,1]^{\Bbb N}$ ...
Boaz Tsaban's user avatar
  • 3,102
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 ...
William B.'s user avatar
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 ...
Boaz Tsaban's user avatar
  • 3,102
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 ...
Robin Saunders's user avatar
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 ...
Jakobian's user avatar
  • 715
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) ...
Alessandro Codenotti's user avatar
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,...
Yly's user avatar
  • 936
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 ...
Daniel Asimov's user avatar
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 ...
JE2912's user avatar
  • 359
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 $W = \{ \text{weak homotopy equivalences} \}$, ...
mathmo's user avatar
  • 161
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 $\{\theta\in\Bbb R^n\mid\|\nabla g(\theta)\|<\eta\}$ ...
金睿楠's user avatar

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