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The standard definition of a topological manifold has at its core that it is locally homeomorphic to $\mathbb{R}^n$ at each point, with some other topological conditions to weed out pathological cases. This obviously works fine practically speaking, and captures the jist of what we want out of manifolds as generalisations of euclidean space (taken topologically).

However, I wanted to know if there was an equivalent, purely topological definition for manifolds also, especially a nice one? The normal definition relies of the structure of the reals for its construction which I find a little displeasing philosophically and aesthetically, given that many spaces in mathematics and physics happen to intrinsically be manifolds without involving the reals at all.

We could of course hack together an equivalent definition by replacing the real line with an equivalent space defined purely topologically, and use that to create a topological stand-in for $\mathbb{R}^n$, but that feels very messy and bad (logically correct, but morally wrong). Very un-insightful.

So, is there a good definition of a manifold, purely in terms of topological primitives?

user6873235
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I am not sure whether this is a complete answer for you, but this is too long for a comment.

A manifold is supposed to be a topological space that locally looks like "the standard space" + maybe some properties to avoid pathologies or to make the theory nicer. There are of course different "standard spaces" depending on the situation, but most often they are $K^n$ for some field $K$ (usually even just $K = \mathbb{R}$ or $K = \mathbb{C}$). Therefore it would be very non-natural to not include one of these fields somehow as you are trying to mimic their behaviour locally. You would have to find some topological property that is equivalent to "locally homeomorphic to $\mathbb{R}^n$. That does not feel very natural in the end if you really have to force one weird property on your space like you said.

If you still want to see an alternative approach of defining various classes of manifolds (like differentiable, smooth etc.), then maybe take a look at the sheaf-theoretic definitions. They provide a (in my opinion) better and more natural description, but you will still see a dependency on $K = \mathbb{R}$ or $K = \mathbb{C}$. In that context you could also take a look at $K$-analytic manifolds for some arbitrary complete valued field $K$. Then you have a quite large class and can still plug in your favorite field depending on the situation. That could probably also work quite well with the different types of manifolds arising in physics (but I have no real knowledge of physics so that is just a guess).

Con
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    I think this is a valuable answer, but not quite the sort of thing I was looking for. I tend to prefer thinking of topology as a kind of generalisation of spatial and geometric concepts (and I consider the soul of $\mathbb{R}$ to be spatio-geometric rather than algebraic) ultimately finding their intellectual ancestry in Euclid and straight-edge and compass constructions, so trying to create a definition which generalises nicely to other fields $K$ feels to me a strange and unnatural condition.

    Thank you nonetheless, I hope your answer is more useful to others.

     – user6873235 Nov 02 '19 at 10:28