Is there a topological characterization of Euclidean spaces?

Question

Suppose $X$ is a topological space. What are the properties such that if $X$ satisfies them, then $X$ is homeomorphic to $\mathbb{R}^{n}$ for some non-negative integer $n$?

There are answers to this for the real line such as here and here. I am wondering if there are similar answers to the case of general Euclidean spaces?

This type of question leaves more than one answer possible, so any suggestions are appreciated.

Answer 1

There is a simpler description, which I discovered after investigating ideas presented by our StackExchange users: Noah Schweber and Taras Banakh. Namely "A compact contractible topological group is trivial" paper by Burkhard Hoffman states:

Corollary 1. Let $G$ be a contractible locally compact topological group. Then $G$ is homeomorphic to a finite product of real lines.

With this we at least get rid of "finite dimension" condition. Although the algebraic aspect is still there.

Answer 2

Following up on one of my comments above, here is a weak (see below) positive answer: responding to a question on MO, Taras Banakh observed that a result of Gleason/Palais implies that every Hausdorff contractible topological space which is the underlying space of some topological group and has finite covering dimension is some $\mathbb{R}^n$.

This characterization has two suboptimal features in my opinion. First is the introduction of algebraic ideas, namely the property of being a "groupizable" space. This could be viewed as a positive, though. Second, and much more seriously in my opinion, is the use of the word "finite" which feels a bit like cheating. Note that if we're allowed to refer to finiteness we could just say "some finite topological power of $\mathbb{R}$" (via an appropriate characterization of $\mathbb{R}$).

Answer 3

The Wikipedia article on inductive dimension notes that a theorem of Urysohn states that when $X$ is a normal space with a countable basis then

LDim X = IDim X = iDim X

(These are in turn, the Lesbegue dimension and the large and small inductive dimension).

And go on to say that the Nobeling-Pontryagin theorem then states if this dimension is finite then they are subspaces of Euclidean spaces.

This gives a purely topological characterisation of subspaces of Euclidean spaces as the real line is not used anywhere here and no algebra is used.