# Questions tagged [gn.general-topology]

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

2,918 questions
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### 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 ...
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### 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 ...
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### 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 $\mathcal{C}(\phi)$...
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### 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 ...
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### 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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### 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 $X$...
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### 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 ...
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### 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 $K$ ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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, ...
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### 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 $\tau = \{\emptyset, X\}$?
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### 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 ...
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### 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....
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### 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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### 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 ...