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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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20 views

Fiber-bundle : continuity of transition maps and inverse in general

Let $(E,\pi,B)$ be a locally trivial fibration, with fiber a topological space $F$, $\Phi_i$ and $\Phi_j$ two trivializations over $U_i$ and $U_j$. The transition map from $i$ to $j$ is the ...
4
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1answer
126 views

Proper homotopy

Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper? I am interested in particular in ...
2
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1answer
665 views

A new generalisation of dimension? part 2

I worked this theory : A new generalization of the dimension? I have a theorem about dimensions which is more general and simple than for matroids. Definition 1: A structure $S$, is a pair $(X, \...
10
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1answer
438 views

Lorenz attractor path-connected?

Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure. EDIT: The answer below is unsatisfactory, and possibly ...
14
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1answer
398 views

The dominating number $\mathfrak{d}$ and convergent sequences

All spaces considered below are compact Hausdorff. If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...
2
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0answers
87 views

What is the boundary of the set $\{ x : dist (x ,\partial \Omega) > \alpha \}$ for a domain $\Omega$?

Let $\Omega$ is a bounded open domain in $\mathbb R ^n$, and $\alpha \geq 0$ a real number, and consider the set $ E_\alpha = \{ x \in \Omega : \text{dist}(x , \partial \Omega) > \alpha\} $, which ...
6
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1answer
121 views

Objects whose morphisms are Lipschitz maps

I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz"). The answer seems pretty clear, and proceeds like the definition of manifolds: 1) If $...
3
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1answer
65 views

Prove that $\mu \left(\left\{t\in X\,;\;\sum_{i=1}^d|\phi_i(t)|^2>r \right\}\right)=0$

Let $(X,\mu)$ be a measure space and $\phi=(\phi_1,\cdots,\phi_d)\in L^{\infty}(X)$. Let $$r=\max\left\{\sum_{i=1}^d|z_i|^2; (z_1,\cdots,z_d)\in \mathcal{C}(\phi)\right\},$$ where $\mathcal{C}(\phi)$...
6
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2answers
307 views

What is the name for a set endowed with a Lipschitz structure?

I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...
6
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1answer
111 views

Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$

The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee_p C\mathbb H$ of two coni over the ...
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1answer
171 views

What is known about topological groups of countable spread in ZFC?

A topological space has countable spread if every discrete subspace is at most countable. By Theorem 8.10 in Todorcevic's book "Partition Problems in Topology", PFA implies that each regular space $X$...
7
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1answer
247 views

Is a Borel image of a Polish space analytic?

A topological space $X$ is called analytic if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$. We say that a topological ...
11
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1answer
385 views

Is the dimension given by Klee trick ever sharp?

The Klee Trick allows one to find an $\mathbb{R}^m$ where two embeddings of same compact metric space have homeomorphic complements. More precisely, given two embeddings of a compact metric space $K$ ...
30
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3answers
840 views

Non embedding of $Y\times Y$ into $\mathbb{R}^3$

I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type. Theorem. Let $Y$ be the union of two segments ...
7
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0answers
227 views

The Klee Trick for subsets of $\mathbb{R}^3$

Update: The lead paragraph has been changed to reflect the solution to a related question. I asked the question Is dimension given by the Klee trick ever sharp? and it has been answered in the ...
7
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0answers
166 views

Topological applications of $\mathfrak{p}=\mathfrak{t}$

I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality. Searching in ...
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0answers
85 views

Topological groups containing the Sorgenfrey line

The Sorgenfrey line $\mathbb S$ is the real line endowed with the topology generated by the base consisting of all half-intervals $[a,b)$ for real numbers $a<b$. The Sorgenfrey line is first-...
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76 views

A special topological space [on hold]

Let $X$ be a compact topological space (not necessarily Hausdorff). I am looking for a charactrization the following property: Property: Every closed subset $C$ of $X$ can be written as a disjoint ...
6
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0answers
106 views

Countable network vs countable Borel network

Definition. A family $\mathcal N$ of subsets of a topological space $X$ is called $\bullet$ a network if for any open set $U\subset X$ and point $x\in U$ there exists a set $N\in\mathcal N$ ...
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1answer
65 views

Is the map between mapping spaces, induced by the functor $\vert Sing(-)\vert$ continuous?

Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibilites, for me it's just important, that I ...
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1answer
493 views

Frankl's conjecture restricted to finite topological spaces

A finite topological space is a finite family of finite sets that is closed under both union and intersection. Frankl's conjecture states that for any finite union-closed family of finite sets, ...
2
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0answers
69 views

A Baire space with meager projections

Question. Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\...
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1answer
73 views

On a pair of continuous functions “connected” by continuous functions

Suppose $X,Y$ are topological spaces with $Y$ homogeneous and $f,g:X\to Y$ continuous such that there exist continuous functions $u,v:Y\to Y$ such that $$f = u\circ g \text{ and } g= v \circ f.$$ ...
13
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1answer
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 \...
0
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1answer
57 views

Maximum of a sum of Gaussian functions

Consider the function which maps $\mathbb{R}^n$ to $\mathbb{R}$ \begin{align} f(x) = \sum_{i=1}^{n} b_i\phi_i(x) \end{align} where $\phi_i(x) = \exp(-\frac{||x-x_i||_2^2}{2})$ are Gaussian functions ...
0
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1answer
60 views

strict topology on multiplier algebras

Suppose $A$ is a $C^*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for ...
6
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0answers
166 views

Does anyone use non-sober topological spaces?

Recall that a sober space is a topological space such that every irreducible closed subset is the closure of exactly one point. Is there any area of mathematics outside of general topology where non-...
3
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1answer
352 views

If non-empty player has a winning strategy in Banach-Mazur game BM(X), then it also has in BM(Y)?

Let $f:X\rightarrow Y$ be a continuous, open, surjection function and second player (non-empty) has a winning strategy (not important which one, say for simplicity stationery st.) in $BM(X)$. Then can ...
2
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0answers
52 views

A possible characterization of stratifiable spaces?

Let us recall that a regular topological space is semi-stratifiable if each point $x\in X$ has a countable family of neighborhoods $(U_n(x))_{n\in\omega}$ such that each closed subset $F\subset X$ is ...
25
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3answers
588 views

What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
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1answer
703 views

Can one determine the dimension of a manifold given its 1-skeleton?

This may be an easy question, but I can't think of the answer at hand. Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...
2
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1answer
72 views

Every $b$-discrete space $X$ with countable injective weight is basically disconnected?

Recall that a space $X$ is called basically disconnected [1] if every cozero-set has an open closure. According to Tkačuk [2], a space $X$ said to be $b$-discrete if every countable subset of $X$ is ...
5
votes
1answer
248 views

Topology of connected subsets of the $3$-torus

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$. We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded. I am ...
4
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0answers
64 views

Is the Baireness a 3-space property of topological groups

It is known that the product of two Baire spaces can be meager. On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...
6
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0answers
59 views

Is each Choquet topological group strong Choquet?

A topological space $X$ is called (strong) Choquet if the player II has a winning strategy in the (strong) Choquet game. It is known that a metrizable space $X$ is $\bullet$ Choquet if and only if ...
14
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1answer
370 views

A parametric version of the Borsuk Ulam theorem

Is there a topological space $X$, which is not a singleton, and satisfies the following property? For every continuous function $f: X\times S^2\to\mathbb{R}^2$ there exist a point $x\in S^2$ such ...
4
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0answers
172 views

homotopy type of box topology.

Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?
10
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1answer
472 views

Homeomorphic characterization of the real line? [duplicate]

Let $A$ be a path-connected subset of $\mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected. Is $A$ necessarily ...
4
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1answer
117 views

What can say about $2^X= \{A\subseteq X: A\text{ is closed set} \}$, when $(X, \mathcal{U})$ is a compact uniform space?

It is known that if $(X, d)$ is a compact metric space, then hyperspace $2^X= \{A\subseteq X: A\text{ is closed set} \}$ is a compact space with Hausdorff metric What can say about $2^X= \{A\...
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3answers
120 views

Extension of continuous map on metric space

Let $X$ be a compact metric space, $A\subset X$ a closed subset and $f:A\to A$ be a continuous map. Can $f$ be extended to a continuous map $X\to X$? If so, is there an extension which is injective if ...
0
votes
1answer
104 views

Smallest collection of linear operators satisfying isometry property

Let $\mathscr{A}=\{(A_{1,1},A_{1,2},A_{1,3}),...,(A_{S,1},A_{S,2},A_{S,3})\}$ be a collection of linear operators $A_{n,k}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. For $u$ and $v\in (\mathbb{R}^2)^3$, ...
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1answer
108 views

First countable geometric realization of a simplicial group

Suppose we have a simplicial group $G$. What do we need from $G$ to get first countable $BG$, where $BG$ is a geometric realization of $G$?
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3answers
320 views

Countable connected space where removing $1$ point destroys connectedness

Is there a countable connected space $(X,\tau)$ such that for all $x\in X$ the space $X\setminus\{x\}$ is not connected any more with the induced subspace topology?
33
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1answer
919 views

Does there exist a continuous 2-to-1 function from the sphere to itself?

I am interested in the following question: Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$? I suspect the answer is no, but I don't know ...
0
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1answer
286 views

Topological properties of complex valued Riemann sum limit curve and a particular integral inequality

I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$): $$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
3
votes
2answers
222 views

Connected topological space $X$ such that $\emptyset, X$ are the only open connected subsets

Let $(X,\tau)$ be connected such that $\emptyset$ and $X$ are the only open connected subsets. Does this imply that $\tau = \{\emptyset, X\}$?
5
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1answer
110 views

Uniqueness of limits and compactness implies closure

It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...
2
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0answers
49 views

Refining monotone-light factorizations

Let $f:X\to Y$ be a continuous map between topological spaces. Consider the quotient map $\pi:X\twoheadrightarrow X/E$ given by decomposing the fibers of $f$ to their connected components. In Lemma 6....
4
votes
1answer
117 views

Is the category of inclusion prespectra bicomplete?

Working in compactly generated weak Hausdorff spaces, is the category of inclusion prespectra bicomplete? I should probably specify that by inclusion prespectra, I mean prespectra such that the ...
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vote
0answers
559 views

A new generalization of the dimension?

During my research, I came a cross on these notions : Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...