# Questions tagged [gn.general-topology]

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

**8**

votes

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240 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

**0**answers

137 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

**3**answers

421 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

**1**answer

152 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

**0**answers

69 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

**0**answers

122 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

**1**answer

116 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

**1**answer

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

**1**answer

102 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 $|D|=|X|$?

**12**

votes

**1**answer

343 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{...

**10**

votes

**1**answer

659 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 ...

**6**

votes

**2**answers

451 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

**1**answer

192 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

**1**answer

404 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

**1**answer

146 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

**2**answers

107 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

**0**answers

143 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

**1**answer

113 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

**1**answer

127 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

**2**answers

88 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

**1**answer

634 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

**1**answer

90 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

**1**answer

186 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 ...

**8**

votes

**1**answer

219 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{...

**3**

votes

**0**answers

52 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

**0**answers

71 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

**1**answer

77 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

**1**answer

92 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 $f^X|_D = f$.
$\...

**6**

votes

**1**answer

289 views

### iterated limit sets of a countable subset of real numbers

Let $A\subset \mathbb{R}$ be a closed subset, and $A'$ be the sets of limit points. We know that if $A$ is a countable set, $A'$ is a proper subset of $A$. Is it possible to find a subset( closed and ...

**6**

votes

**1**answer

217 views

### Is restriction a closed map?

Originally asked on MSE.
Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ ...

**2**

votes

**0**answers

67 views

### Lifting clopen subsets

Let $A$ be a subset of a topological space $T$, we say that clopen subset of $A$ lift to $T$ whenever $L$ is a clopen subset of $A$ the there exists a clopen subset $H$ of $T$ such that $H\cap A=L$.
...

**3**

votes

**0**answers

29 views

### Closedness of the partial order in complete Hausdorff semitopological semilattices

First some definitions.
A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...

**24**

votes

**1**answer

2k views

### Closed balls vs closure of open balls

We work in a separable metric space $(X,d)$. With $\overline{B}(x,r)$ I denote the closed ball around $x$ of radius $r$, and with $cl \ B(x,r)$ I denote the closure of the open ball. Clearly, we ...

**2**

votes

**0**answers

71 views

### Square Peg Problem and curve density

Square Peg Problem (or conjecture) is so famous. See this article
Let $CS:=\{\gamma:S^1\longmapsto\mathbb{R}^2 | \;\;\text {Square Peg Problem is true}\}$ and $C=\{\Upsilon:S^1\longmapsto\mathbb{R}^2 ...

**3**

votes

**1**answer

136 views

### Partitions of unity in constructive mathematics

Can someone point me to any substitutes for the partition of unity in Bishop's constructive mathematics?
In particular, under what circumstances can we construct a partition of unity subordinate to ...

**4**

votes

**3**answers

247 views

### Metrizable subspaces of separable spaces

Are metrizable subspaces of separable spaces separable?
Certainly subspaces of separable metrizable spaces are separable but subspaces of separable spaces need not be separable in general.

**1**

vote

**1**answer

85 views

### Compactness of the Fell topology and local compactness

Given some topological space $\mathbf{X}$, we consider the Fell topology on the set of closed subsets of $\mathbf{X}$. This is generated by sets of the form $I_U = \{A \mid A \cap U \neq \emptyset\}$ ...

**1**

vote

**0**answers

138 views

### Clopen subsets of a closed subspace of a spectral space

Let $X$ be a topological space. Set
$K(X) := \{ A\subseteq X\mid A$ is quasi-compact and open $\}.$ A topological space $X$ is called spectral,
if it satisfies all of the following conditions:
1) $...

**13**

votes

**2**answers

530 views

### Topological obstructions to existence of immersion

Let $M$ be a smooth, non-compact manifold.
a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ?
b) If not, are there some concrete ...

**1**

vote

**1**answer

83 views

### Does this collection of properties imply metrizable?

I am working with a space with the following properties, and want to know if it is necessarily metrizable.
countable union of compact nowhere dense sets
T$_4$$=$T$_1$+normal
separable
Lindelöf
I ran ...

**1**

vote

**0**answers

49 views

### Comparing Different Notions of Unicoherence in the Plane

Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...

**8**

votes

**0**answers

188 views

### Is there a computable homeomorphism between two different Cartesian powers of the computable real numbers?

It's well know that it is surprisingly difficult to prove that $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic for $n\neq m$. Commonly proofs go through Brouwer's fixed point theorem, which is '...

**3**

votes

**1**answer

126 views

### Quotient of compact metrizable space in Hausdorff space

Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...

**1**

vote

**0**answers

88 views

### Dual of $C(X)$ with the compact open topology

Let $X$ be a completely regular space and let $C_k(X)$ be the space of all continuous functions with the compact-open topology. If $X$ is completely metrizable, is the strong dual $C(X)^*$ the strong ...

**6**

votes

**1**answer

151 views

### Can one construct a regular neighborhood without an ambient space?

If I understand my PL topology correctly (and please correct me if I don't), if $K$ is a $k-$complex and $n\ge 2k+2$, then any two PL embeddings $a,b\colon K\to \mathbb{R}^n$ are isotopic. Therefore, ...

**2**

votes

**1**answer

227 views

### What are some surprising facts that happen after you remove a point to a space? [closed]

There are some facts that are really impressive after you remove a point to a space. Some typical examples are the existence of exotic spheres or the fact that
$S^4$ is not almost complex. Or some not ...

**3**

votes

**1**answer

111 views

### Nice representation of open sets in $\sigma$-algebras in certain circumstances

Let $(X,\tau)$ be a topological space. For a given topological base $\mathcal{E}$ for $\tau$, let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$.
Q. Assume ...

**4**

votes

**1**answer

84 views

### Nice arrangement of open sets in $\sigma$-algebras

Let $X$ be a topological space and $\mathcal{E}$ be a topological base for $X$. Let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$.
Q. Let $O$ be an open ...

**1**

vote

**0**answers

64 views

### Does There Exist a Planar, Linear, Triodic Tree-Like Continuum?

Motivated by Continuum image of line is chainable?
A planar continuum $X$ is a compact, connected subset of the plane. It is linear if there is a continuous bijection from $[0,1)$ onto $X$, for ...

**16**

votes

**1**answer

349 views

### Lowest Dimension for Counterexample in Topological Manifold Factorization

Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...