# Questions tagged [homeomorphism]

The homeomorphism tag has no usage guidance.

The homeomorphism tag has no usage guidance.

30
questions

9
votes

1
answer

403
views

Suppose we are given embeddings $f_1,f_2:[0,1]\to\mathbb R^3$.
Does there exist a homeomorphism $g:\mathbb R^3\to\mathbb R^3$ such that $g\circ f_1=f_2$?
This question seems to be classical eighty ...

2
votes

0
answers

69
views

Let $a,b\in \mathbb C$ and define $f_a(z)=e^z+a$ and $f_b(z)=e^z+b$.
Suppose that $f_a$ and $f_b$ have attracting or parabolic cycles (each has a periodic orbit with derivative $\leq 1$). Equivalenty, ...

2
votes

0
answers

91
views

Suppose that $S$ is a (connected) regular surface, possibly with boundary, for instance an annulus. Suppose that both $f$ and $g$ are homeomorphisms whose restriction to $\partial S$ is the identity. ...

1
vote

1
answer

139
views

I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below).
It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be
written as an ...

1
vote

0
answers

69
views

In $C_p$-Theory monographs, it is said that two topological spaces $X$ and $Y$ are said to be $t$-equivalent means that $C_p(X)$ is homeomorphic to $C_p(Y)$. Then they also define $u$-equivalences (...

7
votes

1
answer

245
views

Theorem. Let $\phi:X\rightarrow Y$ be a quasi-isometry between two (Gromov) hyperbolic spaces $X$ and $Y$. If $X$ and $Y$ are proper, then ϕ induces a homeomorphism between their boundaries.
The proof ...

16
votes

2
answers

610
views

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 ...

0
votes

1
answer

74
views

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 ...

1
vote

1
answer

78
views

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 :
$X=\sqcup_{i=1}^{n}U_i$
...

1
vote

0
answers

73
views

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 ...

13
votes

2
answers

949
views

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 ...

1
vote

1
answer

277
views

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 ...

1
vote

1
answer

168
views

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}^...

1
vote

0
answers

81
views

Fix a $d\times d$ matrix $A$, let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous function, and define the induced map $F_{f,A}:\mathbb{R}^d\rightarrow \mathbb{R}^d$ by
$$
x \mapsto \exp(f(x)A)...

1
vote

0
answers

40
views

Let $n>1$ and let $\mathfrak{X}(D_n)$ denote the set of continuous vector fields on the closed disc $D_n\subseteq \mathbb{R}^n$. Let $\operatorname{Homeo}_0(D_n)$ be the set of homeomorphism of ...

3
votes

0
answers

74
views

Let $S^1\subset \mathbb{R}^3$ be the unit circle and suppose $h\colon \mathbb{R}^3\setminus S^1\to \mathbb{R}^3\setminus S^1$ is a homeomorphism. Clearly it might be that $h$ cannot be extended to $S^...

11
votes

1
answer

823
views

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 ...

6
votes

2
answers

358
views

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 ...

0
votes

0
answers

66
views

Is there a characterization of all simply connected, contractable doubling metric spaces which are homeomorphic to a simply connected subset of Euclidean space?

1
vote

2
answers

351
views

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 ...

10
votes

1
answer

197
views

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 ...

9
votes

3
answers

363
views

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely
\begin{equation}
\chi=\...

6
votes

1
answer

356
views

A surjective homeomorphism $h:X\to X$ is minimal if $\overline{\{h^n(x):n\in \mathbb N\}}=X$ for every $x\in X$. In other words, the orbit of each point is dense.
Does either of the Erdös spaces $\...

5
votes

2
answers

426
views

Is the handlebody group of genus two surface generated by Dehn twists along properly embedded disks and annuli?
Are there alternative ways to describe a set of generators that are conceptually simple ...

4
votes

0
answers

183
views

To clarify, I'm speaking of homeomorphisms in a graph theoretic context, defined by subdivisions of arcs in a directed graph. A subdivision of an arc $(x,z)$ in a directed graph is obtained by ...

1
vote

0
answers

81
views

Let $X$ be a regular CW complex, $\sigma$ an $n$-dimensional cell of $X$ and $\tau$ an $(n-1)$-dimensional face of $\sigma$.
Is it true that the pair $(\bar\sigma, \bar\tau)$ is homeomorphic to the ...

10
votes

3
answers

393
views

Let $X$ be a locally compact Hausdorff space. Call $X$ rigid if its only autohomeomorphism is the identity, $\operatorname{Homeo}(X)=\{1\}$.
Questions:
Let $X$ be rigid. Is it true that every open ...

10
votes

2
answers

492
views

Let $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ be the groups of homeomorphic and Borel automorphisms of a space $M$, respectively.
Question: Are $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ ...

6
votes

1
answer

508
views

I would like to get a demonstration of the stable homeomorphism conjecture (SHC$_n$), stating that any orientation-preserving homeomorphism $\mathbb{R}^n \rightarrow \mathbb{R}^n$ is stable in ...

12
votes

3
answers

519
views

Let $A, B \subset S^2$ be closed, countable sets and $\phi \colon A \rightarrow B$ be a homeomorphism. Can we extend $\phi$ to a homeomorphism from $S^2$ to itself?
It is well-known that the answer ...