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The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. Aesthetically, this seems kind of ugly to me. The real line is a high-tech piece of mathematical machinery. We build up all that structure, then build the definition of a manifold out of it, then throw away most of the structure. It seems kind of like building an airplane by taking a tank, adding wings, and getting rid of the armor and the gun turret.

I've spent some time trying to figure out a definition that would better suit my delicate sensibilities, and have come up with the following sketch: An $n$-dimensional manifold is a completely normal, second-countable, locally connected topological space that has Lebesgue covering dimension $n$, is a homogeneous space under its own homeomorphism group, and is a complete uniform space.

Does this work? I should reveal at this point that I'm a physicist, and no more than a pathetic dilettante at math. I have never had a formal course in topology. My check on my proposed definition consisted of buying a copy of Steen's Counterexamples in Topology and flipping through it to try to find examples that would invalidate my proposed definition.

Since I'm not competent as a mathematician, what would probably be the best outcome of this question would be if someone could point me to a book or paper in which my idea is carried out by someone competent.

Clarification: I mean a topological manifold, not a smooth manifold.

Also, I should have mentioned in my original post that I had located some literature on the $n=1$ case in terms of characterizing the real line (which is not, of course, the same as characterizing a 1-dimensional manifold, but is a related idea):

P.M. Rice, "A topological characterization of the real numbers," 1969

S.P. Franklin and G.V. Krishnarao, "On the Topological Characterization of the Real Line: An Addendum," J. London Math Soc (2) 3 (1971) 392.

Brouwer, "On the topological characterization of the real line," http://repository.cwi.nl/search/fullrecord.php?publnr=7215

Kleiber, "A topological characterization of the real numbers," J. London Math Soc, (2) 7 (1973) 199

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    Lang informally referred to (differentiable) manifolds as open subsets of Banach spaces being glued together by $C^p$-differentiable mappings (which identify parts of these subsets). So I think it would make more sense to improve this definition (using sheaves maybe, I don't know) rather than trying to come up with something that effectively obscures local homeomorphism to a Banach space.2011-07-22
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    @Alexei: But when you say "Banach space", you implicitly mean "Banach space over $\mathbb{R}$", don't you? I'm pretty sure a space that is locally homeomorphic to a Banach space over $\mathbb{Q}_p$ cannot be a topological manifold.2011-07-22
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    Your delicate sensibility is going to have to prove that (assuming that list of properties does actually characterize manifolds) that "your manifolds" are the same as "ours" at some point or another... In a similar vein, even though we *know* that a Lie group is simply a topological group whose identity element has a neighborhood which does not contain a subgroup, no one would sanely use that as a definition, because to go from there to, well, Lie groups takes enormous effort.2011-07-22
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    The long line is a perfectly good 1-dimensional manifold that isn’t second countable.2011-07-22
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    To be honest and to play with your analogy (heehee!), I think you're trying to get a tank, strip it off, rebuild it without using the screws, and then still call it a tank in some manner.2011-07-22
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    @Brian M. Scott: is it? I believe that in general one defines a manifold $M$ to be Hausdorff, second countable and loc. hom. to $\mathbb{R}^n.$ If there is some reason why people do not demand second countability, I'd love to hear about it.2011-07-22
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    @Zev: at first I thought you were nitpicking, but actually this may be a good idea to consider p-adic manifolds (defined as you put it) in their own right :)2011-07-22
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    What structure of $\mathbb R$ do we throw away by making the common definition of a manifold?2011-07-22
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    I think this is nice question. I myself wondered about similar things in many different contexts. E.g. being path-connected requires $\mathbb R$ too and this also seems like an overkill (for finite spaces, say). Also, there is another question lurking close: what makes $\mathbb R$ so special that we use to build so many structures when (naively) many other objects could stand in its place. It seems to me that the answer will be that it's just what people had done and they found it's ok for their purposes. Yet, one can surely ponder replacing reals with something else.2011-07-22
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    This is completely irrelevant to the actual topic at hand, but I just really, _really_ wanted to link to [http://en.wikipedia.org/wiki/Winged_tank](http://en.wikipedia.org/wiki/Winged_tank).2011-07-22
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    What I don't like about this approach is that it might even lead to a purely topological definition of a topological manifold (and *that* may be considered as interesting), but I don't see how you're eventually going to differentiate, speak of tangent bundles etc in this framework.2011-07-22
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    @Alexei: Actually, I meant to be nitpicking (sorry :)). The OP is looking for a definition of manifold without reference to the reals; I don't think your proposal meets that criterion.2011-07-22
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    @Gerben: There is a considerable literature on non-metrizable manifolds; see the survey by Peter Nyikos in the *Handbook of Set-Theoretic Topology*, K. Kunen and J.E. Vaughen, eds. (A manifold is second countable iff it is Lindelöf iff it is paracompact iff it is metrizable.)2011-07-22
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    @Brian: thanks, I'll have a look.2011-07-23
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    "The real line is a high-tech piece of mathematical machinery." - and a manifold isn't ?2011-08-07
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    I like this question very much. At the same time, I think that the the "extra structure of the real line" may be more relevant than the "winged tank" metaphor suggests. Think of building a manifold using the "real line plus extra structure" like building a skyscraper using "steel girders plus scaffolding": the scaffolding may be removed from the final structure, but it makes it much easier to get all the other components in place.2011-08-08
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    If one could give a strictly topological definition / characterization of manifolds, this in turn would give us a strictly topological definition / characterization of $\mathbb{R}$ as the unique non-compact 1-manifold. Would it assuage your sensibilities to find such a characterization? Then the definition of manifolds that we have now --- nice (2nd countable, Hausdorff) and locally an $n$-fold product of $\mathbb{R}$---would carry through, except that we never had the field structure on $\mathbb{R}$ to start with.2011-08-11

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