# Questions tagged [gn.general-topology]

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

3,717
questions

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### Does there exist a regular $P$-space which is strongly star-Lindelof but not star-Menger?

A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset ...

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61 views

### Chains and disjoint families of sets

The following construction appears in a paper (from a top statistics journal) that has confounded me: Given a family of compact, disjoint sets $\mathcal{K}=\{K_1,\ldots,K_r\}$, the author considers a ...

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156 views

### What is known about differentiable and analytic structures on the long line (and half-line)?

When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ ...

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62 views

### Covering map preserved under homotopy equivalence

Given a $m-$sheeted covering map from $p:M^n\to N^n$, where $M,N$ are manifolds of dimension $n$. Suppose $M$ and $N$ are homotopy equivalent to finite CW complexes $X$ and $Y$ of same dimension $k$. ...

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54 views

### Two sheeted covering with torus as total space [closed]

Let $X \to Y$ be a $2-$sheeted connected covering, where $X$ is $S^1 \times S^1$. What are the choices of $Y$? I know at least $S^1\times S^1$ and Klein Bottle are two choices. Are there any other ...

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48 views

### Euler characteristic of pseudomanifolds with boundary

It is a well-known fact that for every compact oriented odd-dimensional manifold $\mathcal{M}$ with boundary it holds that
$$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M}).$$
In particular, if ...

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58 views

### Conditions guaranteeing density of Hölder functions in $C(X,Y)$

Let $X$ and $Y$ be Polish metric spaces with $X$ compact. Let $C^{\alpha}(X,Y)$ denote the set of $\alpha$-Hölder functions from $X$ to $Y$ (for $\alpha \in (0,1]$) and let $C(X,Y)$ denote the set of ...

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266 views

### The closure of the set of injective continuous functions

Setup/Notation:
Let $n,m\in \mathbb{N}$ and let $C(\mathbb{R}^n,\mathbb{R}^m)$ be the space of continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ equipped with the compact-open topology. Let $...

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68 views

### Is the singular value decomposition a measurable function?

$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators
$$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$
where $\mathbb U_n$ is the ...

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126 views

### Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?

Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that
$$\...

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203 views

### Group structure on the strip

Let $X$ is a strip between two different parallel lines $a$ and $b$ on a plane ($a,b\subset X$) and $h(x)=\min\limits_{l\in \{a,b\}}\{d(x,l)\}$.
Let $(X,*)$ be a topological group with the following ...

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135 views

### How to check a fiber bundle is trivial

Given a smooth fiber bundle $X \to S^1,$ such that the fiber, $F$, is homotopic to $S^2 \vee S^2.$ Is it true that this is always a trivial fiber bundle?
In general, how to check a fiber bundle is ...

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137 views

### When does $C(X)$, $X$ a continuum, admit a continuous choice function?

Given a continuum $X$ (compact metrizable connected $X$) let $K(X)$ denote the hyperspace of nonempty compact subspaces of $X$ with the Vietoris topology and let $C(X)$ denote the (closed) subspace of ...

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155 views

### Does every open set contain a dense $F_{\sigma}$ subset?

Let $U$ be a regular open set in a Tychonoff space $X$ (regular means that it is an interior of a closed set).
[ In my specific situation $U$ is of the form $\operatorname{int} f^{-1}(0)$, where $f$ ...

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451 views

### VC dimension of standard topology on the reals

Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is an open set $U$ with $D=S\cap U$?
I'm asking merely out of curiosity, but I'll mention that ...

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195 views

### A strong Borel selection theorem for equivalence relations

In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16):
Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...

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102 views

### On ultraweak continuity [closed]

Let $A$ be a C*-algebra, $f$ be a representation of $A$, $F$ be the universal representation of $A$, and $g=f \circ F^{-1}$. For an ultraweakly continuous linear functional $w$ on $f(A)$, $w\circ g$ ...

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201 views

### A question about locally compact spaces

Recently I read a book about linear algebraic group written by Ian Macdonald. There is a conclusion which I can't prove.
It says that if $X$ is locally compact Hausdorff space, then $X$ is compact if ...

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830 views

### Why it is convenient to be cartesian closed for a category of spaces?

In 1967 Steenrod wrote what later became a quite celebrated paper, A convenient category of topological spaces (Michigan Math. J. 14 (1967) 133–152). The paper conveys the work of many (among the most ...

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143 views

### Can Tychonoffs theorem for a countable number of spaces be proven with ZF plus the axiom of (countable) dependent choice?

It can be proven without any form of infinite choice that the product of two compact spaces (and thus any finite product) is compact, while on the other hand, it is well known that the general form of ...

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80 views

### Are open subsets of a $\sigma$-compact LCH space $\mathcal{K}$-analytic?

I'm reading Guedj and Zeriahi's Degenerate Complex Monge-Ampère Equations Chapter 4 which talks about capacities. Specifically Corollary 4.13 claims that when $X$ is a locally compact Hausdorff $\...

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193 views

### Is there a notion of „flatness” in point-set topology?

In algebraic geometry, flat morphisms are usually associated with the intuition that they have „continuously varying fibers”. Is there a notion in topology formalizing the same intuition? Consider for ...

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129 views

### How can one construct this dendrite?

In the early 1970s Pelczynski noticed that the only surjective isometries on $C(K)$ for the following compact Hausdorff space $K$ are $\pm Id$. I believe this was the first such example.
Quoting from ...

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52 views

### Does there exist a strongly star-Lindelöf space which is not DCCC?

A space $X$ is said to be strongly star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $A$ of $X$ such that $St(A,\mathcal U)=X$.
A space $X$ has discrete ...

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**1**answer

140 views

### Understanding Kelley's intersection number (Boolean algebras)

It is known that:
Theorem (Kelley, 1959). There exists a finite, strictly positive, finitely additive measure on a Boolean algebra $A$ if and only if $A^+$ is the union of a countable number of ...

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191 views

### Is the derivative the unique operation on points in the plane that preserves convexity?

Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$
Such that the ...

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101 views

### Inclusion inducing isomorphism at all level except one

Let $V$ is a projective hypersurface of dimension $3$ and $D$ be divisor at infinity of $V$ (assume $D$ has isolated singularities). It is known that the third homology of both $V$ and $D$ are hard to ...

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69 views

### Pareto-optimal front $F$ in $m$-dimensional space can not have more than $\mathbf{H}_{m-2}(F)$ homology groups

I need to prove that a Pareto-optimal front $F$ in $m$-dimensional space (i.e. $m > 1$) can not have more than $\mathbf{H}_{m-2}(F)$ homology groups.
What it simply means that in a 2-dimensional ...

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127 views

### About uniform continuity

Is there a definition Df(g) of uniform continuity of g, without using the notion of metric?
Let $(E,d_E)$ and $(F, d_F)$ metrics spaces, $f$ continuous fonction of $E$ to $F$
We must have :
Df$(f)$ ...

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116 views

### Continuity of Kakutani fixed points

Let $X$ be a compact and convex space and let $T=[0,1]$ be some parameter space. Let $F:X\times T\rightrightarrows X$ be a correspondence that is compact-valued, convex, and upper-hemicontinous. By ...

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975 views

### Can $[0,1]^4$ be partitioned into copies of $(0,1)^3$?

Is there a partition of $[0,1]^4$ such that every member of the partition is homeomorphic to $(0,1)^3$?
More generally, I would like to know for which values of $m$ and $n$ there is a partition of $[0,...

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384 views

### Are the “topologies” arising from constructive type theories with quotients actually condensed sets?

This is the second in a pair of questions. For the other see Are representations in computable analysis the equivalent to countably-generated condensed sets?.
Dustin Clausen and Peter Scholze have a ...

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329 views

### Are representations in computable analysis the equivalent to countably-generated condensed sets?

This is the first in a pair of questions. For the other see here.
Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, ...

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125 views

### Compactness of symmetric power of a compact space

Suppose I have a compact metric space $(X,d)$ and let $\mathcal{X}=X^K$ be the product space. Consider the equivalence relation $\sim$ on $\mathcal{X}$ given as: for $\alpha,\beta\in \mathcal{X}$, $\...

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87 views

### A closed subset of a Dedekind-complete order has subspace topology equal to order topology

Here's a fairly easy fact from point-set topology that I'm having trouble finding a reference for. Say $X$ is a total order satisfying the least-upper bound property, and $S$ is a closed subset of it....

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177 views

### Reference request: large generalized probability measures

I'm interested in references relevant to the following: what is the right generalization, if there is one, of a probability measure that takes on values in an structure of more than continuum size?
I'...

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130 views

### Uniquely selecting points from open pairwise disjoint refinements of an open cover

Let $\mathcal U$ be an open cover of some space $X$.
Let $\{\mathcal V_\alpha:\alpha<\kappa\}$ enumerate all of its pairwise-disjoint
open refinements.
When is it possible to define sets $Z_\alpha$ ...

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55 views

### Existence of a proper Morse function

I started with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded (local embedding) it inside $R^{2n}$. Now take a regular neighbourhood $U$ of $X$ in $R^{2n}$ which has the same homotopy ...

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84 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 ...

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84 views

### Necessary and sufficient conditions for the Lie group embedding $G \supset J$ can be lifted to $G$'s covering space [closed]

Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so
$$
G \supset J.
$$
If $G$ has a nontrivial first homotopy group $\pi_1(G) \neq 0$.
If $G$ has a universal cover $\widetilde{G}$, ...

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883 views

### Sequences with 0's in $\mathbb R ^\omega$

Let $\mathbb R ^\omega$ be the set of all sequences of real numbers in the product topology.
Let $X$ be the set of all sequences in $\mathbb R ^\omega$ which have at least one 0.
Let $Y$ be the set of ...

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**1**answer

69 views

### Existence of a Hölder homeomorphism satisfying prescribed norm constraints

Let $\Omega$ be a convex body$^{\boldsymbol{1}}$ in $\mathbb{R}^n$ where $n$ is a positive integer. Fix a positive integer $k$ and some $0<\alpha\leq 1$. Let $k_1> k_2>0$. Does there ...

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209 views

### A $*$-homomorphism $C(X) \to C(Y)$ gives a continuous map $Y \to X$

Given a $C^*$-algebra $A$, we write $\Omega(A)$ for its space of characters, i.e. its non-zero algebra homomorphisms $A \to \mathbb{C}$. If $X$ is a compact Hausdorff space, it is well-known that
$$X \...

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583 views

### How can I construct a closed manifold from a finite CW complex?

If I start with a, say, 3-CW complex $X$ which can be embedded in $\mathbb{R}^5$, I can get a neighbourhood $U$ of $X$ which has the same homotopy type of $X$. Then $U$ is a $5-$ dimensional open ...

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977 views

### A topological version of the Lowenheim-Skolem number

This is a continuation of an MSE question which received a partial answer (see below).
Given a topological space $\mathcal{X}$, let $C(\mathcal{X})$ be the ring of real-valued continuous functions on $...

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328 views

### Does the functor $\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$ have an adjoint?

Consider the category $\mathbf{Top}$ of topological spaces, the category $\mathbf{Topos}$ of toposes and geometric morphisms, and the category $\mathbf{Loc}$ of locales. Let
$$\mathrm{Sh}\colon\mathbf{...

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84 views

### Complex manifold structure on a real manifold of real dimension 2n

A real manifold of dimension $2n $ sometimes can be given a complex structure. If I start with a $3-$dimensinal CW complex $X$ which can be embedded in $\mathbb{R}^6$ and get a neighbourhood of $X,$ ...

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**2**answers

172 views

### Intersection of all open subgroups vs. the intersection of all open normal subgroups

I am interested to know examples of topological groups $G$ for which the intersection $\bigcap\{H\leq G\mid H\text{ open}\}$ of all open subgroups of $G$ is the trivial subgroup but for which the ...

**7**

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658 views

### Finite CW complex with finite non-abelian fundamental group and higher homologies zero

I want to build a finite CW complex such that $\pi_1$ is non-abelian and $H_i$ are zero for $i\geq 2.$
From Hatcher for a given group G, one can create an example of a 2-complex $X_G$ with $\pi_1(X_G)=...

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77 views

### Subsets of $\mathbb{R}$, every nonempty subset of which generates a disconnected translation-invariant topology

Let $\mathbb{R}$ be the set of real numbers. Given a subset $S$ of $\mathbb{R}$, let $\mathcal{T}_S$ be the translation-invariant topology generated by $S$. That is, $\mathcal{T}_S$ is the topology ...