What topological properties uniquely characterize the local topology of $\mathbb{R}^n$?
I know $\mathbb{R}$ is the unique complete ordered field, but that's partly algebraic.
Better if the characterization avoids sneaking in the topology of $\mathbb{R}$ through the side door by saying it has a metric which is continuous, for example. And certainly we don't want to sneak it in by assuming the notion of a manifold.
A start: a similar question on MathStack, and a 1965 paper giving an axiomatic characterization of $S^n$. (Chapter II of the paper is the meat.)
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$\begingroup$You are actually quite mistaken that algebraic topology is irrelevant in regards to your question. But the math involved in it is quite deep, here are some glimpses.
Definition 1. A finite-dimensional locally compact metrizable topological space $X$ is called ANR (absolute neighborhood retract) if it is locally contractible.
Definition 2. An (integer) $n$-dimensional homology manifold is a metrizable 2nd countable topological space $X$ satisfying$$ H_*(X, X-\{x\}) \cong H_*({\mathbb R}^n, {\mathbb R}^n \setminus \{0\}) $$for every $x\in X$. (Due to the excision, this condition is actually local.)
Definition 3. A generalized manifold is an ANR which is also a homology manifold.
All topological $n$-dimensional manifolds are generalized manifolds. It is a well-known open problem to find extra topological conditions under which the converse is also true. For instance, if $X$ is a 2-dimensional generalized manifold, then it is also a topological surface. This is a nontrivial theorem due to Moore (1920s). From this, with a bit more work, one gets a characterization of 2-dimensional disks. (One needs to define generalized manifolds "with boundary." Then the 2-disk is the compact contractible 2-dimensional generalized manifold with boundary.)
In higher dimensions ($n\ge 3$), Moore's result fails and one needs extra conditions. One such condition (which kicks in once $n\ge 5$) is known as DDP (the Disjoint Disk Property):
Definition 4. A metrizable topological space is said to satisfy the DDP if any two (continuous) maps of the 2-disk into $X$ admit arbitrarily small perturbations, which have disjoint images.
DDP fails for all manifolds of dimension $\le 4$ but holds for all manifolds of dimension $\ge 5$. For a long time, it was believed that every generalized manifold of dimension $\ge 5$, that also satisfies DDP, is, in fact, a topological manifold. This, turned out to be false:
Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S., Topology of homology manifolds, Ann. Math. (2) 143, No. 3, 435-467 (1996). ZBL0867.57016.
Nevertheless, there is an extra invariant, introduced by Frank Quinn (of local algebro-topological nature) of generalized manifolds $X$, whose vanishing (plus DDP) is equivalent to the the condition that $X$ is a topological manifold provided that $n\ge 5$ (see the reference in the above paper). But I stop here.
$\endgroup$ 2 $\begingroup$This is a difficult question, and I don’t think it’s solved. An easier problem is to characterize the topological spaces that embed as a connected closed subset of some Euclidean space.
The necessary and sufficient topological properties are:$(1)$ connected; $(2)$ compact; $(3)$ second countable; and $(4)$ of finite dimension (large inductive, small inductive, covering, they’re all the same in this setting). If you fix $n=1$ then this amounts to conditions $(1,2,3)$, plus having at most two noncut points (so you’ve got to be either degenerate or an arc). In the case $n=2$, there are all kinds of partial characterizations of continua that live in the plane.
A very elegant theorem is due to Kuratowski: A connected topological graph is planar if it does not contain either the complete graph on five vertices or the $3–3$ bipartite graph. As for some necessary conditions vis a vis your question: If $ X$ is a connected open subset of $\mathbb{R}^n$ then $X$ is: $(1)$ path connected; $(2)$ locally path-connected; $(3)$ second countable; and $(4)$ of dimension $\le n$ .
There may be lots of other interesting topological properties I’ve forgotten; one interesting feature of our space $X$ is that if $f:X→\mathbb{R}^n$ is continuous and one-one, then $f$ is a homeomorphism onto $f(X)$ and $f(X)$ is also open in $\mathbb{R}^n$. (This is from Brouwer’s invariance of domain theorem, $1912$.)
Sorry if this doesn't answer your question but I hope it helps.
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