# Questions tagged [gn.general-topology]

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

4,052
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### Topological rings with a final topology

Given a family of ring homomorphisms $ \phi_i : X \rightarrow Y_i $ where each $ Y_i $ is a topological ring and consider the initial topology on $ X $, i.e. the coarest topology such that each map is ...

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### Is there any example of a Lindelof space that has no Menger dense subspaces?

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

3
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+50

### $\Sigma_*$-product is not $\sigma$-countably compact

In Arhangel'skii's book "Topological function spaces" there is a part where the author uses that, if $\kappa>\omega$ is a cardinal number, then the space $$\Sigma_*(\kappa):=\left\{x\in \...

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### Is the domain space in Lusin's theorem required to be Hausdorff?

I'm reading a general version of Lusin's theorem, i.e.,
If $\mu$ is a finite Radon measure on $X$, and $Y$ is a second countable topological spaces, then for any Borel-measurable function $f:X\to Y$ ...

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### Is each typical map on the $n$-cube strongly rigid?

This question is inspired by this (still unanswered) MO-post.
A function $f:X\to Y$ between topological spaces is called strongly rigid if every continuous self-map $h:X\to X$ with $f\circ h=f$ is the ...

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64
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### Second homology group of a presentation complex

I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the questions are silly.
Given a finite group $G$, and a presentation $P$ of ...

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1
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120
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### Formula for the Euler characteristic of a local system on $\mathbb{P}^1$

Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...

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### Are closed embeddings characterized by a left lifting property in the category of topological spaces?

It is well-known and easy to check that a continuous map between topological spaces is an embedding if and only if it has the LLP with respect to $A \to *$ and $B \to *$ where $A$ is the two-point ...

4
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1
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### Given $f$ from the cylinder $C$ to the interval constant on one boundary, is there a $r:C\to C$ constant on a boundary with $f\circ r = f$?

My question might be trivial, but my lack of knowledge of this particular subject has not enabled me to find the answer. What I want to know is the following. Let $I=[0,1]$ and $C=S^1\times I$ be the ...

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1
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### Name of a space with both a topology and a metric that are not compatible?

Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.
Is there a canonical name for such a structure (maybe ...

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### Is there a condensation of a closed subset of $\kappa^\omega$ onto $\kappa^\omega\setminus A$ …?

Let $\aleph_1\le\kappa<c$ and $A\subset \kappa^{\omega}$ such that $\lvert A\rvert\le\kappa$.
Is there a condensation (i.e. a bijective continuous mapping) of a closed subset of $\kappa^\omega$ ...

4
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### Is each TS-topologizable group TG-topologizable?

Definition 1. A topology $\tau$ on a group $X$ is called
$\bullet$ a semigroup topology if the multiplication $X\times X\to X$, $(x,y)\mapsto xy$, is continuous in the topology $\tau$;
$\bullet$ a ...

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2
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176
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### Function space and contractibility

$\DeclareMathOperator\map{map}$I have the following question:
Let $X$ and $Y$ be topological spaces. Let $\map(X,Y)$ denote the space of non-constant continuous functions from $X$ to $Y$. Suppose ...

3
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2
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361
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### Regularity of lipschitz and derivable function

Let be lipschitz $f$ on $[0,1]$ and everywhere derivable. Is it true that $f\in C^1([0,1])$ ?

3
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### Topologically embed Klein bottle into $\mathbb{R}^4$ projecting to usual "beer-bottle" surface in $\mathbb{R}^3$

(Originally asked in 2018 at https://math.stackexchange.com/questions/2946505/topological-embedding-of-klein-bottle-into-mathbbr4-that-projects-to-usual?noredirect=1#comment9514257_2946505;cross-...

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### On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$

Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$.
For which $X$'s is it true that $J_A+J_B=J_{A\cap ...

2
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0
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43
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### Can we decompose an increasing net of functions into two increasing nets with prescribed supports?

Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...

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### Continuity of "inversion operator" between function spaces

Question: When is the operation of inversion continuous as a map between spaces of invertible functions?
Let $\mathcal{F}$ be a function space such that $f\in\mathcal{F}\implies$ $f$ is invertible and ...

2
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0
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### Separating property of a finite union of topological disks

Let $X$ be a topological $2$-sphere. Let $D_1, D_2, \dots, D_n \subset X$ be a finite family of closed topological disks (i.e. sets homeomorphic to the closed unit disk). Let $\mathcal{U} = \bigcup_{1 ...

2
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1
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198
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### The Borel sigma-algebra of a product of two topological spaces

The following problem arose while trying to justify some "known results" in abstract harmonic analysis on noncommutative groups, for which I couldn't find explicit statements in the ...

7
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### State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye :
"If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...

3
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1
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645
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### Condensed mathematics

I have a little technical question on Peter Scholze's lectures on condensed mathematics.
On page 12, right above the Proof of Theorem 2.2, he says that for extremally disconnected sets the condition (...

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2
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342
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### Do these irrationals exist?

An irrational $a$ verifies : $\{a\times n+k;(n,k)\in\mathbb Z^2 \}$ is dense in $\mathbb R$.
If you take $a$ universe then : $\forall b\in \mathbb N^*, \{a\times n^{b}+k;(n,k)\in\mathbb Z^2\}=A(a,b)$ ...

2
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2
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412
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### How to use that the Hessian is negative definite in this proof

Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...

4
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1
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84
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### A neighborhood of a 2-disc $D\subset\Bbb R^4$ that tapers off towards the boundary?

I am given a PL 2-disc $D\subset\Bbb R^4$ (everything PL from here on) and I need a "neighborhood" $N\simeq B^4$ (PL-homeomorphic to a 4-ball) so that $\partial N\cap D=\partial D$.
If I got ...

3
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1
answer

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### Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$.
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of ...

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0
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### Density of Lipschitz functions in Bochner space with bounded support

Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...

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1
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147
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### If $M$ is contractible manifold and $X\subset \partial M$, does the cone over $X$ embed in $M$?

Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C_X$ the cone over $X$.
Question: Is it true that $C_X$ embeds in $M$ with its boundary $\partial C_X$ mapped to $X\subset \...

2
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0
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56
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### Length metrics on covering spaces

This is a question (Exercise 3.30(2)) in the book `Metric spaces of non-positive curvature' written by Bridson and Haefliger.
In the book, there is the following proposition (Proposition 3.28)
Let $p:\...

2
votes

1
answer

247
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### (Dis)prove : if every function with closed graph are continuous then the target space is compact

$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological spaces.
$\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $.
Question : Does this implies $(Y, \tau_Y) $ is compact?
...

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### $n$-connected spaces (terminology)

A graph is called $n$-connected if it remains connected after removal $\le n$ vertices.
Question. What is the name of an analogous property of topological spaces: a space that remains connected after ...

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### What is the name for a point that is periodic to within $\varepsilon$?

Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is ...

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### Two topologies on the space of maps from an algebraically closed field to a projective variety

This question is related to this one but I have written this in a self-contained manner.
All varieties are complex varieties.
For quasi-projective variety $U$ and a projective variety $X$ we can ...

6
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148
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### What is a non-smooth connection?

Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $\...

7
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2
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308
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### How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?

I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere.
Here $\Omega$ is any ...

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### Properties on morphism of locally convex vector spaces

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $U,V,W,W'$ $K$-vector spaces, such that $U$ is a Banach-space and $W,W'$ are finite dimensional. Further we have an (algebraic) short exact ...

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### Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?

Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...

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1
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147
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### Tensor product is complete?

Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...

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### In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?

I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...

2
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1
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### Subsets of $\mathbb{S}^n$ fixed by an orientation-reversing self-homeomorphism — Part 2

This is another special case of this question.
Recall that we call a subset $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing homeomorphism $h: \mathbb{S}^n \to \mathbb{...

2
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### Subsets of $\mathbb{S}^n$ fixed by an orientation-reversing self-homeomorphism — Part 1

Call a subset $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing self-homeomorphism $h: \mathbb{S}^n \to \mathbb{S}^n$ fixing $Z$ pointwise.
Question 1: Which subsets of ...

3
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### An extension of Stone duality

First let me recall Stone duality in terms of propositional logic.
Let $L$ and $K$ be propositional signatures (i.e., sets of propositional variables). Let $T$ be a propositional theory over $L$ and $...

10
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1
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460
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### Stone–Čech compactification as a semigroup

Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...

3
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### Whether a functional which preserves maximum for comonotone functions is monotone?

Let $X$ be a compactum (compact Hausdorff space). By $C(X,[0,1])$ we denote the space of continuous functions endowed with the sup-norm We also consider the natural lattice operations $\vee$ and $\...

2
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1
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74
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### Is there a bound on the number of connected components of a zero set of an integrable function?

If $f$ is a real-analytic function on $[0,1]^n$, and $f$ has finite differential transcendence degree, is there some way to bound the number of connected components of its zero set or the set where it ...

6
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1
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179
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### Steinhaus number of a group

$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$.
Let $\mathcal A_X$ be the family of ...

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0
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64
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### Morphism in commutative square strict?

Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism.
Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...

3
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0
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29
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### Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup

A semigroup $X$ endowed with a topology is called
$\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous;
$\bullet$ a semitopological semigroup if for every $a,b\...

0
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2
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108
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### Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)

Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
\mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...

0
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0
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72
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### Rothberger property and semi-open sets

Here is the definition of a space $X$ to be Rothberger, if for each sequence $(\mathcal{U}_{n})_{n\in\mathbb{N}}$ of open covers of $X$, there exists a sequence $(U_{n})_{n\in\mathbb{N}}$ where $U_{n}\...