# 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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**1**answer

352 views

### Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?

Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\...

**6**

votes

**0**answers

94 views

### Spatiality of products of locally compact locales

In Johnstone´s Sketches of an Elephant Volume 2, page 716,
lemma 4.1.8 states that for spatial locales $X$ and $Y$ with $X$ locally compact then the locale product $X\times Y$ is spatial.
Is this ...

**2**

votes

**1**answer

98 views

### Direct proof a property of hyperstonean spaces

First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...

**3**

votes

**1**answer

70 views

### Does each separator between points of a continuum contain an irreducible separator?

Definition. A closed subset $S$ of a topological space $X$ is called a separator between points $x,y\in X\setminus S$ if the points $x$ and $y$ belong to different connected components of $X\setminus ...

**4**

votes

**1**answer

131 views

### The homological negligibility of certain subsets in compact manifolds

Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).
I need a reference to the following facts (which I believe are true at least in dimension $n=3$):
Fact 1. For every ...

**-4**

votes

**1**answer

204 views

### Borromean rings, Condorcet's paradox and Quantum chromodynamics [closed]

In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in ...

**3**

votes

**0**answers

43 views

### Irreducible separators of compact manifolds

Definition. A closed subset $S$ of a topological space $X$ is called
$\bullet$ a separator of $X$ if $X\setminus S$ is disconnected;
$\bullet$ an irreducible separator if $S$ is a separator of $X$ ...

**9**

votes

**1**answer

227 views

### Set of homeomorphic fixed points that is dense, but not equal to whole space

If $(X,\tau)$ is a topological space, let $FH(X)$ denote the collection of $x\in X$ such that there is a non-identity homeomorphism $\varphi:X\to X$ with $\varphi(x) = x$.
What is an example of a $...

**6**

votes

**2**answers

146 views

### Thrice intersecting closed geodesic on genus 2 orientable closed surface

Does there exist a closed geodesic on a closed genus 2 orientable surface (with hyperbolic metric) that self-intersects at only one point thrice?

**13**

votes

**1**answer

356 views

### Limit of homeomorphisms from square to square

Let $\square=[0,1]\times[0,1]$ be the unit square
and $f\colon\square\to \square$ is a continuous map that fixes the points on the boundary.
Assume $f$ is a limit of homeomorphisms $\square\to \...

**9**

votes

**1**answer

252 views

### Topological dimension of $p$-adic manifolds

What is the topological dimension of a (locally analytic) $p$-adic manifold over a non Archimedean field $K$?
Is the topological dimension of $K^n$, $n$?

**1**

vote

**0**answers

88 views

### Path connected without bounded path connected subset?

Question: Is there a path connected subset of $\mathbb R^2$, without any bounded path connected subset (aside from singletons)?
Motivation: If we replace "path connected" by "connected", then the ...

**4**

votes

**2**answers

181 views

### Unramified map of Riemann surfaces

Let $f:S \to T$ be a surjective, unramified, holomorphic map between connected Riemann surfaces. If $S$ is not compact is it always true that $f$ is a covering?
This is of course true if $S$ is ...

**6**

votes

**1**answer

175 views

### Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?

Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a ...

**6**

votes

**1**answer

124 views

### Analogue of Urysohn metrization for Lawvere metric spaces?

Urysohn proved that any regular, Hausdorff, second-countable space $X$ is metrizable, i.e. there exists a metric space whose underlying topological space is $X$. But what if we ask the same question ...

**8**

votes

**1**answer

164 views

### Is $\beta\mathbb N$ a unique compactification with the smallest possible permutation group?

For a compactification $c\mathbb N$ of $\mathbb N$ let $\mathcal H(c\mathbb N,\mathbb N)$ be the group of homeomorphisms $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\...

**5**

votes

**1**answer

216 views

### Is each compactification of $\mathbb N$ soft?

Definition. A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be soft if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\...

**0**

votes

**1**answer

92 views

### Continuity of maps in which preimage preserves compactness

Let $X$ and $Y$ be Hausdorff spaces and suppose that $Y$ is locally compact. Let $f:X\to Y$ be a surjective map such that for any compact subset $K \subset Y$ the pre-image $$f^{-1}(K)=\{x\in X: f(x)\...

**6**

votes

**1**answer

185 views

### Tools for constructing homeomorphisms between 4-manifolds

(I am a complete amateur in topology, so this is a question out of curiosity.)
The question was inspired by this post Fake versus Exotic . What methods can, realistically, be used to construct a ...

**3**

votes

**0**answers

60 views

### Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found ...

**13**

votes

**1**answer

513 views

### Bijection $f: \mathbb R^n \to \mathbb R^n$ that maps connected onto connected sets must map closed connected onto closed connected sets?

Willie Wong asked here (MO) and here (MSE) very interesting question.
As he phrased it:
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$...

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votes

**1**answer

173 views

### Component and quasi-component

Let $X$ be a topological space and $x\in X$. Then the quasi-component of the point $x$, denoted by $C_x$, is the intersection of all clopen (closed-and-open) subsets of $X$ which contain the point $x$...

**1**

vote

**1**answer

92 views

### Does regular $G_\delta$ imply normal?

I'm trying to prove that if every closed set in a topological space is regular $G_\delta$, then the space is normal. By regular $G_\delta$, I mean for any closed set $A$,
there exists a countable ...

**5**

votes

**2**answers

230 views

### Complete atomless Boolean algebras with abelian automorphism group

Is there any example of a complete atomless Boolean algebra with a non-trivial abelian automorphism group?
This is equivalent, by Stone duality, to asking for an extremally disconnected compact ...

**1**

vote

**1**answer

104 views

### More general metric spaces (where image of metric is not a subset of $\mathbb R$)

When defining metric spaces we want that $d$ in a pair $(X,d)$ satisfies:
1) $d$ is a function from $X \times X$ to $\mathbb R$
2) $d(x,y) \geq 0$ with $d(x,y)=0$ iff $x=y$
3) $d(x,y)=d(y,x)$
4) $...

**1**

vote

**1**answer

69 views

### Approximate Selection for finite-valued Upper Hemicontinuous/Semicontinuous Maps?

I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...

**2**

votes

**0**answers

327 views

### A property of subspaces of a topological space

Let $X$ be a topological space and $A$ be a subset of $X$. Is there any nice condition on $A$ equivalent to the property that if $C$ is a connected component of $X$, then either $A\cap C=\emptyset$ or ...

**3**

votes

**1**answer

211 views

### If $X, Y$ are topological spaces, with $Y$ being a k-space, and $f : X \to Y$ is a proper covering map, is $X$ necessarily a k-space?

A k-space is a compactly generated Hausdorff topological space. (I used the terminology "k-space" in the question, in order keep the question within the limit of 150 characters.)
Note that under the ...

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vote

**1**answer

167 views

### Group action on quasi-isometric geodesic metric space [closed]

If a group $G$ acts on a geodesic metric space $X$, then does $G$ act on a geodesic metric space $Y$ which is quasi-isometric to $X$?

**3**

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69 views

### Is $\Box_{n\in\omega} \mathbb{R}$ metacompact?

Is $\Box_{n\in\omega} \mathbb{R}$ (that is $\mathbb{R}^\omega$ endowed with the box topology) metacompact?

**5**

votes

**1**answer

207 views

### On filters possessing a countable network

Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$
A family $\mathcal N$ of subsets of $\omega$ is called a network ...

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vote

**1**answer

43 views

### Complexity of set of fibers on which a set is relatively clopen

Let $X$ and $Y$ be compact metrizable spaces with $f:Y\rightarrow X$ an open surjection. Suppose that $G\subseteq Y$ is a closed set. How topologically complicated can the set $\{x\in X : f^{-1}(x)\...

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vote

**1**answer

83 views

### Quantity of partition sets intersecting a compact set

Let $X$ be a compact metric space.
Let $\{X_\alpha:\alpha\lt \mathfrak c\}$ be a partition of $X$ into $\mathfrak c=|\mathbb R|$ dense first category $F_\sigma$-subsets of $X$.
Let $A$ be a non-...

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vote

**4**answers

240 views

### Is a separable compact Hausdorff space already metrizable? [closed]

It is a known fact that a 2nd countable compact Hausdorff space is metrizable. What if we weaken the 2nd countable to separable only - is the space still metrizable?
The core of the question, or a ...

**3**

votes

**0**answers

79 views

### Is each metric continuum $\ell_p$-chain connected?

This problem was motivated by the MO problems:
"Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?".
...

**4**

votes

**1**answer

122 views

### For any $n \in \Bbb N$ , does there exist $A \subset \Bbb R$ such that $A^1,A^2,\ldots,A^{n-1}$ are non-empty and $A^n = \emptyset$?

In a topological space $X$ , $a$ is defined to be a condensation point of a set $A$ in $X$ if and only if each neighborhood of $a$ meets $A$ in uncountably many points.
Let $A^c$ denote the set of ...

**6**

votes

**1**answer

119 views

### Are $\varepsilon$-connected components dense?

Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of ...

**9**

votes

**1**answer

277 views

### Is every metric continuum almost path-connected?

The question was motivated by this question of Anton Petrunin.
By a metric continuum we understand a connected compact metric space.
Let $p$ be a positive real number. A metric continuum $X$ is ...

**0**

votes

**0**answers

73 views

### A finiteness property of profinite sets

I would like to understand the canonical topology on the category of profinite sets. Unless I am making mistakes, this translates to the following question in point set topology:
Say $X$ is a ...

**10**

votes

**0**answers

114 views

### Characterizing compact Hausdorff spaces whose all subsets are Borel

I am interested in characterizing compact topological spaces all of whose subsets are Borel. In this respect I have the following
Conjecture. For a compact Hausdorff space $X$ the following ...

**8**

votes

**1**answer

691 views

### Interpreting a space in Baire space: how many facts do I need to understand the whole thing?

Below I'm working in ZF+DC+AD or similar; I want enough choice that things don't explode, but I also want the Wadge hierarchy to be well-behaved everywhere. Since this question is a bit long, I've put ...

**8**

votes

**0**answers

197 views

### Universally meager spaces and large cardinals

Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...

**1**

vote

**0**answers

91 views

### Determine all possible magnetic monopole of gauge theories

In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact:
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It ...

**10**

votes

**1**answer

183 views

### homeomorphisms induced by composant rotations in the solenoid

Let $S$ be the dyadic solenoid.
Let $x\in S$, and let $X$ be the union of all arcs (homeomorphic copies of $[0,1]$) in $S$ containing $x$.
$X$ is called a composant of $S$.
It is well-known ...

**12**

votes

**1**answer

368 views

### When can I “draw” a topology in Baire space?

The motivation for this question is a bit convoluted, so in the interests of conciseness I'm just asking it as a curiosity (and I do find it interesting on its own); if anyone is interested, feel free ...

**7**

votes

**1**answer

440 views

### Space filling curve whose all level sets are finite (countable)

Is there a continuous surjective function $f:[0,1] \to [0,1]^2$ such that
every level set $f^{-1}(y)$ is a finite set? If the answer is no, what about if we replace the finiteness of level sets by "...

**2**

votes

**1**answer

117 views

### Topologically Ordered Families of Disjoint Cantor Sets in $I$?

Suppose that we have an uncountable collection $C_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is ...

**28**

votes

**1**answer

608 views

### Running most of the time in a connected set

Let $P$ be a compact connected set in the plane and $x,y\in P$.
Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small?
...

**4**

votes

**1**answer

159 views

### Exponential law w.r.t. compact-open topology

It is well-known that if a topological space $Y$ is
locally compact (not necessarily Hausdorff),
then the map
$$
\operatorname{Hom}(X \times Y, Z) \to
\operatorname{Hom}(X, Z^Y)
$$
(here we use the ...

**18**

votes

**1**answer

427 views

### Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?

Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$?
Remark 1. The answer to the ...