Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

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

3
votes
0answers
51 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 ...
-1
votes
0answers
41 views

Approximation by regular sets

Let $(X,\Sigma,\mu)$ be a Radon space and $X$ be a compact metric space. By definition of $\mu$, every Borel set $A\in \Sigma$ can be approximated from within by compact sets. However, if $A$ is ...
1
vote
1answer
35 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)\...
1
vote
1answer
71 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-...
1
vote
4answers
175 views

Is a separable compact Hausdorff space already metrizable?

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 ...
2
votes
0answers
50 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
1answer
106 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
1answer
106 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
1answer
184 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
0answers
64 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 ...
8
votes
0answers
88 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 ...
6
votes
1answer
180 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
0answers
174 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
0answers
78 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 ...
7
votes
1answer
144 views
+100

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 ...
9
votes
1answer
303 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
1answer
420 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
1answer
112 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 ...
27
votes
1answer
559 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
1answer
139 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 ...
15
votes
1answer
369 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 ...
8
votes
2answers
219 views

Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?

According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point. However, these fixed points cannot be chosen ...
2
votes
0answers
106 views

When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is a cross-post to the question I asked at MSE. Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...
12
votes
3answers
391 views

Can the real line be embedded in a space $X$ such that all the nonempty open subsets of $X$ are homeomorphic?

The question is in the title: Q1: Is there a topological space $X$ containing a copy of the real line and having the property that all the nonempty open subsets of $X$ are homeomorphic? Let us ...
5
votes
1answer
142 views

“König's theorem” for $T_2$-spaces?

For any topological space $(X,\tau)$ we define a matching to be a collection of non-empty and pairwise disjoint open sets. We define the matching number $\nu(X,\tau)$ to be the smallest cardinal $\...
0
votes
0answers
67 views

Is there a standard definition for this topology setup?

There are two complete metric spaces $(A,d_A)$ and $(B,d_B)$, also $B \subset A$, so there is a third induced metric space $(B,d_A)$. There is a continuous and onto function $e:A\to B$. For any $b \in ...
3
votes
0answers
107 views

Identification of ultrafilters with measures

We know that each ultrafilter $p$ on $\mathbb{N}$ can be identified with a finitely additive $\{0,1\}$-valued probablity measure $\mu_{p}$ on the power set of $\mathbb{N}$. Now my question is which ...
2
votes
1answer
106 views

Path Metric Topology

Is there an example of a metric space $(X,d)$ whose corresponding path metric, $d^\prime$ generates a strictly finer topology compared to the topology generated by $d$?
1
vote
1answer
81 views

Maxed-out Hausdorff metric

Let $(Y,d)$ be a non-degenerate compact metric space, and let $d_H$ be the Hausdorff metric (https://en.wikipedia.org/wiki/Hausdorff_distance) on $K(Y)$ generated by $d$. Here $K(Y)$ is the set of ...
3
votes
1answer
98 views

Connected spaces where every dense set is large

Let $\kappa >\aleph_0$ be a cardinal. Is there a connected space $(X,\tau)$ with $|X| = \kappa$ such that for every dense set $D\subseteq X$ we have $|D|=|X|$?
-2
votes
0answers
49 views

Some questions aboute a generalization of connectedness

Let $A$ be a subset of a topological space $X$. We say that $A$ is a clopen lifting subset of $X$ if whenever $L$ is a clopen subset of $A$ then there exists a clopen subset $H$ of $X$ such that $H\...
12
votes
1answer
290 views

Obstruction of spin-c structure and the generalized Wu manifods

Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the $$ H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
9
votes
1answer
563 views

Is every complete Boolean algebra isomorphic to the quotient of a powerset algebra?

Is every complete Boolean algebra isomorphic to a quotient, as a Boolean algebra, of some powerset algebra $\wp(X)$? It is not true for arbitrary Boolean algebras, see the comments, or see my MathSE ...
5
votes
2answers
402 views

Any 3-manifold can be realized as the boundary of a 4-manifold

We know "Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
11
votes
1answer
177 views

If $G$ is a paracompact topological group, then is $G \times G$ paracompact?

If $G$ is a paracompact topological group, then is $G \times G$ paracompact? This question is raised by Gepner and Henriques (first paragraph of 2.2). Of course, this is not true for arbitrary ...
9
votes
1answer
375 views

On the Large Cardinal Strength of Normal Moore Space Conjecture

In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces: Theorem. (Jones) If $2^{\aleph_0}<2^{\aleph_1}$ then all separable normal Moore spaces are metrizable. Then ...
6
votes
1answer
127 views

Reference request: A collection of topologies on $\mathbb{N}$ formed via series

First, some quick notation: for any series $\sum_{n=1}^\infty a_n$ whose terms are positive real numbers, and for any subset $M = \{m_1, m_2,...\} \subseteq \mathbb{N}$, we write $\sum_M a_n$ to mean ...
3
votes
2answers
99 views

What is the dimension of a subspace of the product of $n$ linearly ordered compacta

This question is motivated by this problem of Dominic van der Zypen. Problem. Let $X=\prod_{i=1}^nX_i$ be the Tychonoff product of linearly ordered compact Hausdorff spaces $X_1,\dots,X_n$. Is it ...
11
votes
0answers
137 views

A connected Borel subgroup of the plane

It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...
0
votes
1answer
99 views

When every open cover admits a $\sigma$-disjoint subcover?

We say that a sequence $(\mathcal X_n)$ of families of subsets of a topological space $X$ is a $\sigma$-disjoint cover of $X$ if every family $\mathcal X_n$ consists of mutually disjoint sets and $\...
4
votes
1answer
120 views

Order convergence vs topological convergence in partially ordered sets

Short version of the question. If $(P,\leq)$ is a partially ordered set (poset), a topology denoted by $\tau_o(P)$ can be defined (see below). There is also another notion of convergence, called order-...
4
votes
2answers
87 views

A reasonable topology on the group of minimal usco maps

An usco map is an abreviation for an upper semicontinuous multi-valued map with non-empty compact values. An usco map $f:X\multimap \mathbb R$ is called minimal is it coincides with each usco map $g:X\...
14
votes
1answer
610 views

Homotopy pullback of a homotopy pushout is a homotopy pushout

Let's assume that we have a cube of spaces such that everything commutes up to homotopy. The following holds: - The right square is a homotopy pushout and - all the squares in the middle are ...
1
vote
1answer
77 views

Gluing locally defined continous functions over complex domain

This is a cross-post to the question I asked at MSE over almost a month ago. Suppose $n, l, m \in \mathbb N$ and $n \ge l > m$. Let $T: \mathbb C \to \mathcal M(n \times l; \mathbb C)$ be ...
1
vote
1answer
180 views

Question on K.Gobel's paper 1969

Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K $, with $K$ a nonempty, closed, convex, bounded subset ...
7
votes
1answer
207 views

Question about taking the Zariski closure in $\mathbb{A}_{\mathbb{R}}^n$

Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with the Zariski topology, where closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials. Suppose $V \subseteq \mathbb{...
2
votes
0answers
46 views

(When) is the Dedekind-MacNeille completion of a po-set Hausdorff?

Let $X$ be a p.o. Consider the topology on $X$ generated by $$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$ Throughout this discussion I shall refer to ...
1
vote
0answers
69 views

Is there any characterization for lifting clopen subsets

Let $Y$ be a subset of a topological space $X$. We say that a clopen subset $L$ of $Y$ lifts to $X$ whenever there exists a clopen subset $H$ of $X$ such that $H\cap Y=L$. Let $X$ be a compact and $...
1
vote
1answer
60 views

Borel $\sigma$-algebra on the space of Hölder continuous functions

Let $(M,d)$ be a separable metric space $E$ be a $\mathbb R$-Banach space $\alpha\in(0,1]$ Moreover, let $$\left\|f\right\|_{C^{0+\alpha}(K,\:E)}:=\sup_{x\in K}\left\|f(x)\right\|_E+\sup_{\substack{...
4
votes
1answer
91 views

Are homogeneous $T_2$-spaces flexible?

We say that a topological space $(X,\tau)$ is flexible, if for every closed discrete subset $D\subseteq X$ and every map $f: D\to X$ there is a continous map $f^X:X\to X$ such that $f^X|_D = f$. $\...