# Questions tagged [homeomorphism]

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### What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?

If $M$ is a smooth connected closed $n$-dimensional manifold, its universal covering space is homeomorphic to Euclidean space $R^{n}$, and its fundamental group is $Z^{n}$, then is it homeomorphic ...
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### Existence of a Hölder homeomorphism satisfying prescribed norm constraints

Let $\Omega$ be a convex body$^{\boldsymbol{1}}$ in $\mathbb{R}^n$ where $n$ is a positive integer. Fix a positive integer $k$ and some $0<\alpha\leq 1$. Let $k_1> k_2>0$. Does there ...
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### Preserving the Holomorphicity of a Complex Differentiable Form on a Polytope

I had originally intended the following to be a secondary question to my original post but then realized that it merited a separate entry entirely. Question: Could it be possible to approximate a ...
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### Disjoint union of clopen sets such that the fibers has constant cardinality [closed]

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that : $X=\sqcup_{i=1}^{n}U_i$ ...
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### Existence of global section, étale map and totally disconnected space

I am trying to show the following result : Let $Y$ be a totally disconnected space and compact space, $X$ a locally compact space and $p:Y\to X$ a surjective local homeomorphism. Then, there exist ...
885 views

### Elementary proof that $\mathbb{R}^3 \setminus \{p_1,\dots,p_n\}$ is not homeomorphic to $\mathbb{R}^3$

I was wondering if there were a proof of the fact that $$\mathbb{R}^3 \setminus \{p_1,\dots,p_n\} \: \text{is not homeomorphic to} \: \mathbb{R}^3$$ for every $n \geq 1$ that does not use cohomology ...
223 views

### Extension of homeomorphisms

Let $f,g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be smooth injective and let $n\leq m$. Let $k \in \mathbb{N}$, and let $\iota_m^{m+k}:\mathbb{R}^m\rightarrow \mathbb{R}^{m+k}$ be the canonical ...
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### Why are homeomorphism groups important?

For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
329 views

### Do mixing homeomorphisms on continua have positive entropy?

I am trying to understand relations between various measures of topological complexity. I have read that expansive homeomorphisms on continua, for example, have positive entropy. But I do not know ...
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### Contractable and Simply Connected Doubling Spaces Homeomorphic to Euclidean Space

Is there a characterization of all simply connected, contractable doubling metric spaces which are homeomorphic to a simply connected subset of Euclidean space?
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### Extending homeomorphisms between compact metric subsets

Let $X$ be a compact metric, second countable space with finite covering dimension. Let $A,B$ be two closed subsets of $X$. Assume that $h:A\to B$ is a homeomorphism. Is it possible to extend $h$ to a ...
190 views

### homeomorphisms induced by composant rotations in the solenoid

Let $S$ be the dyadic solenoid. Let $x\in S$, and let $X$ be the union of all arcs (homeomorphic copies of $[0,1]$) in $S$ containing $x$. $X$ is called a composant of $S$. It is well-known ...