What topological properties need a space have to be topologically euclidean?

Question

To clarify, if I have some some topological space, what are the sufficient and necessary topological properties it must have to be topologically identical to a euclidean space of given dimension?

I have only a very loose, piecemeal understanding of topology currently, so please be light with the jargon, or give explanations. If possible, define things in terms of open sets or neighborhoods, I've found that definitions of closed sets/closure/interior have a nasty habit of either being circular or so dryly axiomatic that I have trouble intuiting it, so I don't understand them well currently.

Answer 1

There is actually quite a lot of theory on such characterisations. It regularly happens that we have a nice standard topological space $S$ (like $\mathbb{R}^n, \mathbb{Q}$, the Cantor set etc.) and that we can draw up a list of topological properties $\mathscr{L}$ such that a space $X$ if homeomorphic to $S$ iff $X$ satisfies all properties from $\mathscr{L}$. These are called topological characcterisations of $S$. We like the list to be as small and intuitive as possible, of course, and not all spaces admit such a nice characterisation (already for the reason there are at most countably many finite lists of "properties", and way more spaces), but since the start of topology as a field quite a few have been shown.

For the real line $\mathbb{R}$ this is known, the list is:

• connected.
• locally connected.
• $T_3$ (i.e. regular and $T_1$).
• separable.
• Every point of the space is a strong cut point. (a strong cut point of $X$ is such that $X$ with that point removed has exactly two components.)

I'm not aware of such a list for $\mathbb{R}^n$ for $n > 1$. A trivial or self-circular property ("$X$ is homeomorphic to the plane" would qualify as a topological property, though a boring one in this context) is of course not allowed. It could be that topological dimension could play a role; this is a topologically invariant way to assign a natural number or $\infty$ to a space, such that $\mathbb{R}^n$ gets assigned $n$ for all $n$. There are several such definitions. I think the case for a general Euclidean space is open (it's not mentioned in Engelking as being solved, nor in any of my other books).

Other spaces are simpler: e.g. $\mathbb{Q}$:

• countable.
• first countable.
• $T_3$.
• no isolated points.

The Cantor set:

• compact.
• second countable.
• $T_3$.
• basis of clopen subsets.
• no isolated points.

But both of these are very disconnected. It seems easier to prove characterisations of such spaces. Only in infinite-dimensional spaces do we get positive results again. (e.g. all separable metric complete infinite-dimensional linear spaces are homeomorphic, so we get fewer "types" of spaces).