# Questions tagged [gn.general-topology]

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

3,421
questions

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23 views

### Relationship between tangent cone and support of density operators

Let $H$ be a separable Hilbert space. Let $(T,\Vert \cdot \Vert_1)$ be the Banach space of self-adjoint trace-class operators on $H$ with the trace norm $\Vert \cdot \Vert_1$. Finally, let $D$ be the ...

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**1**answer

59 views

### Proving neighborhood of a compact product space contains a sub-neighborhood formed by taking product [closed]

I am self studying basic topology and have trouble proving the following question.
If $A$ and $B$ are compact, and if $W$ is a neighborhood of $A \times B$ in $X \times Y$, find a neighborhood $U$ of ...

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votes

**1**answer

304 views

### What is Bouziani space and what are its applications in mathematics?

I have accrossed a new topological space seems were derived from Hilbert Space and it used to solve some boundary value problem for PDE and ODE , Inspired by this paper (page 4, Definition 3.1) , The ...

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133 views

### Is every Baire metric space a complete metric space in disguise?

I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here:
Let's say that a metric space $X$ is Baire if every countable ...

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51 views

### A weakly sequentially continuous operator which is not weakly continuous

I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.
So, let
$T$ an operator between a Banach space $X$ and itself.
$T$ is weakly ...

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98 views

### Products of cones and cones of joins

The join of $A$ and $B$ is the pushout of the diagram
$$
CA \times B \gets A\times B \to A\times CB,
$$
which can be formulated in either the pointed or unpointed topological
category. This pushout is ...

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87 views

### On Čech-complete space

I'm reading an article of topology and i came across a Properties :
Properties :
Closed subspaces and arbitrary products of Čech-complete spaces are Čech-complete
Every Čech-complete space is a Baire ...

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votes

**1**answer

214 views

### The space of skew-symmetric orthogonal matrices

Let $M_n \subseteq SO(2n)$ be the set of real $2n \times 2n$ matrices $J$ satisfying $J + J^{T} = 0$ and $J J^T = I$. Equivalently, these are the linear transformations such that, for all $x \in \...

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**1**answer

82 views

### Covering of discrete probability measures

Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...

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**1**answer

63 views

### Continuous selection parameterizing discrete measures

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...

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297 views

### Homotopical characterization of CW complexes

Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$.
Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...

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169 views

### Homotopical characterization of manifolds

Let $X$ be a compact metrizable topological space of covering dimension $4$.
Assume that for any point $x\in X$ any neighbourhood of $x$ contains a contractible open neighbourhood $U$ such that $U\...

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34 views

### A local base for space of probability measures with Prohorov metric

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...

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vote

**1**answer

47 views

### irreducible compact set vs prime compact set

Let $X$ be a topological space.
A compact set $K$ is called irreducible if for any two compact subsets $K_1$, $K_2$ of $K$ with $K$ is equal to the union of $K_1$ and $K_2$, then $K$ is equal to $K_1$ ...

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61 views

### Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in $n$ variables and up to degree $d$)

Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector ...

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139 views

### Fundamental group and countability

A friend of mine on a Discord server talked about an exercise she had to do (she's in master's degree): prove that you can put an uncountable number of disjoint "5" in the euclidean plane $\...

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votes

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4k views

### What are good mathematical models for spider webs?

Sometimes I see spider webs in very complex surroundings, like in the middle of twigs in a tree or in a bush. I keep thinking “if you understand the spider web, you understand the space around it”. ...

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309 views

### Axiom of Countable Choice and meager sets

Let us recall that the Axiom of Countable Choice (denoted by ACC) says that the countable product $\prod_{n\in\omega}X_n$ of nonempty sets $X_n$ is nonempty.
It is easy to see that ACC implies that ...

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73 views

### Is the space of metric topologies over a given set dense (in the order sense)?

Suppose that $S$ is an infinite set and that $\alpha$ and $\beta$ are metrics over $S$ such that the topology induced by $\alpha$ is everywhere strictly finer than the metric induced by $\beta$, ...

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91 views

### Complement of contractible locally Euclidean subspace

Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?

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137 views

### Homotopy groups of ball complement

Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$.
Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...

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147 views

### Does there exist a complete metric space which is Rothberger (or Menger) but not Hurewicz?

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

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60 views

### A finite union of convex bounded open subsets of $S^{n}$ has finitely generated cohomology groups [migrated]

Let $V_{1},...,V_{n}\subset S^{d}$ be convex, bounded, open subsets of $S^{d}$. Show that $H^{d-1}(\cup_{i=1}^{n} V_{i})$ is finitely generated.
According to a solution here Complement of a finite ...

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151 views

### Sierpinski's characterization of $F_{\sigma\delta}$ spaces

According to [2]: Let $X$ be a space. We call a system $(X_s)_{s\in T}$ a Sierpinski
stratification of $X$ if $T$ is a nonempty tree over a countable alphabet and $X_s$ is a closed subset of $X$ for ...

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**1**answer

123 views

### Function series of normal lower semi-continuous functions

For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is
defined as follows:
$
f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in
U\right\} :U\in ...

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47 views

### Are these boolean subalgebras always the clopen sets of some topology?

What would one call a family $\mathcal{F}$ of subsets $X$ such that:
$$(1):X\in\mathcal{F}$$
$$(2):A,B\in\mathcal{F}\implies A\setminus B\in\mathcal{F}$$
$$(3):P\subseteq\mathcal{F}\text{ is a ...

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votes

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3k views

### An intelligent ant living on a torus or sphere – Does it have a universal way to find out?

I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...

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votes

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241 views

### Non-sequential spaces in the wild

TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...

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vote

**1**answer

125 views

### Regularity and ultrafilter

I read the following result in an article.
Let $X$ be a regular space. Let $\mathcal{M}$ be free closed
ultrafilter on $X$. Set $\mathcal{U=}\left\{ U:U\text{ is open and there
exists a }F\in \mathcal{...

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**1**answer

139 views

### P-filter property?

Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega_i$ where $\omega_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows
$(\bigsqcup_i ...

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votes

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153 views

### Nonhomeomophic spaces with homeomorphic mapping cones

It is natural to ask if it is possible for the mapping cone $X\cup_\alpha CA$
to be homeomorphic to the mapping cone $X\cup_\beta CB$ with $A$ and $B$
nonhomeomorphic. Is there a standard go-to ...

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vote

**1**answer

139 views

### Is every homeomorphism approximately a product of homeomorphisms?

Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...

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126 views

### Set of null-sequences is not $\sigma$-compact

I am interested in a reference for the following fact (or a similar result).
PROPOSITION. Let $X$ denote the set of real null sequences; i.e., the set of $(a_n)_{n=0}^{\infty}$ with $a_n\to 0$, with ...

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**1**answer

136 views

### Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?

Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...

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93 views

### Problem of Thickening an Arc in a Topological $ 2 $-Manifold

Let $ M $ be a topological $ 2 $-manifold (possibly with boundary), $ C $ an arc in the interior of $ M $ (i.e., an injective continuous function from $ [- 1,1] $ into $ \operatorname{Int}(M) $), and $...

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70 views

### Compact metrizable contractible locally contractible topological space of finite covering dimension is a CW complex

Let $X$ be a compact metrizable contractible locally contractible topological space of finite covering dimension. Is $X$ homeomorphic to a CW complex?

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vote

**1**answer

50 views

### Projecting Graph of a Function acted on by a homeomorphism

Let $X,Y$ be compact, connected, simply-connected, and separable, metric spaces each with at-least $2$-points, and let $f,g:X\rightarrow Y$ be continuous functions. Does there always exist a ...

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**1**answer

308 views

### Any continuous map is homotopic to one assuming fixed values at finitely many points

Let $X$ and $Y$ be topological spaces. Assume $X$ is locally contractible and has no dense finite subset. Assume $Y$ is path-connected.
Given $n$ pairs of points $(x_i, y_i)$ where $x_i\in X$ and $y_i\...

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122 views

### adding one point from the Stone-Cech compactification

Let $X$ be any non-compact Tychonoff space and $\beta X$ be its Stone-Čech compactification.
The following fact is known: any point $p$ from the reminder $\beta X \setminus X$ is not a $G_{\delta}$-...

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56 views

### Separating a certain planar region with an open set

I have a fairly specific question related to plane separation properties. I couldn't quite see how to use Phragmen–Brouwer properties to answer it because those kind of results generally apply to ...

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159 views

### When every closed and connected subset is path connected

Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?

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169 views

### Intrinsically defining smooth/continuous/analytic functions

In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing ...

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**1**answer

114 views

### Are there minimal topological conditions on a space 𝑋 for it to have a countable separating set?

Are there minimal topological conditions on a space $X$ for it to have a countable separating set?
A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions ...

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102 views

### 'Monodromy' for relative homology group

Let $A$ and $X$ be topological manifolds. Denote by $\mathbb {Emb}(A,X)$ the space of all topological embeddings $A\to X$.
A loop $f_s:A\to X$ ($s\in[0,1]$) in $\mathbb{Emb}(A,X)$ should give rise to ...

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**1**answer

56 views

### Equicontinuity-like property of a convex compact set

Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...

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votes

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124 views

### Local cross-sections for free actions of finite groups

Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...

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**1**answer

260 views

### Mapping $\mathbb P$ onto $\mathbb Q ^\omega$

Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G_\delta$-...

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**1**answer

230 views

### Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request

I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...

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42 views

### Critical Growth of Dimension for Dense Cover by Linear Subspaces

Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...

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**1**answer

147 views

### Arc connectedness of product spaces

Is arc connected-ness well-behaved with respect to products?
That is -
$\prod X_\alpha$ is arc connected iff $X_\alpha$ is arc connected $\forall \alpha$
In this question on MathStackexchange, an ...