Spaces that are locally homeomorphic to themselves

Question

We know that, for every point $p$ of $\mathbb R^n$, there is a fundamental system of neighbourhoods (namely disks) of $p$ which open sets are all homeomorphic to the whole space (in this case, $\mathbb R^n$).

Now what about topological spaces in general? Manifolds are locally homeomorphic to $\mathbb R^n$, but perhaps for some of them there exist special fundamental systems (different from charts) that satisfy the condition. This doesn't work for $S^n$, for example, because a small neighboorhood of a point is contained in some $\mathbb R^n$, and $S^n$ cannot be embedded in $\mathbb R^n$.

Is my condition false for all manifolds not homeomorphic to $\mathbb R^n$? And what about topological spaces that are not manifolds?

Answer 1

Here's a collection of $n$-manifolds with this property, namely, any open subset $M \subset \mathbb{R}^n$. And lots of these are not homeomorphic to $\mathbb{R}^n$, for example $M = \{x \in \mathbb{R}^n \,|\, x \ne 0\}$.

For the proof, given $p \in M$, start with a fundamental system of neighborhoods of $p$ of the form $$U_1 \supset U_2 \supset \cdots $$ such that each $U_i$ is homeomorphic to $\mathbb{R}^n$.

Set $i_1=1$. Now proceed inductively. Inside $U_{i_k}$ there is an open set $V_{k}$ containing $p$ and homeomorphic to $M$: simply choose some homemorphism $f_{k} : \mathbb{R}^n \to U_{i_k}$ such that $f_k(p)=p$ and then define $V_k = f_k(M)$. Inside $V_{k}$ there exists some $U_{i_{k+1}}$.

The sequence $$V_1 \supset V_2 \supset \cdots $$ is a fundamental system of neighborhoods of $p$ all homeomorphic to $M$.

And, come to think about it, this describes all $n$-manifolds having this property, namely, those manifolds homeomorphic to an open subset of $\mathbb{R}^n$.

Answer 2

A lot of classical homogeneous spaces obey this property:

$\mathbb{Q}$ does, as all countable metric spaces without isolated points are homeomorphic to the rationals. So all open intervals in $\mathbb{Q}$ are too.

$\mathbb{P}$, the irrationals as a subspace of the reals, as it is homeomorphic to $\mathbb{N}^\mathbb{N}$ and so are basic open subsets of that space.

The Cantor set has a clopen local base at any point of neighbourhoods homeomorphic to the Cantor set.

$n$-Manifolds $X$ need not have this property: they have local neighbourhoods that are homeomorphic to $\mathbb{R}^n$, but not necessarily to $X$ (which is what you ask for). So this fails for the circle, and also other spheres.