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Most definitions I've seen for a manifold are based on the idea that small neighborhoods are homeomorphic to [itex]\mathbb{R}^n[/itex]. To me this feels a little like defining a bicycle as a car that's missing the engine and both the wheels on one side. The real number system is this big, sophisticated piece of mathematical technology that includes all kinds of metrical apparatus, but the whole idea of a manifold is that it's nonmetrical. I've been trying to work out a definition of a manifold that avoids this unaesthetic feature. I'm not intent on reinventing the wheel, so if someone says "I know a definition that doesn't invoke the reals, it's on page x of book y," that would be great. But anyway, here's what I've come up with so far:

A manifold is a Hausdorff, first-countable topological space T such that:

(M1) Self-similarity: Given any two points in T, any sufficiently small open neighborhoods around these points are homeomorphic.

(M2) Density: T is completely metrizable.

(M3) Not fractal: T's Hausdorff dimension and Lebesgue covering dimension are equal.

([EDIT] This set of axioms has problems. #23 is an attempt at fixing them.)

I intend M1 to rule out things like a a manifold with boundary, or a line glued onto a plane, and M2 to rule out things like the rationals.

Does anyone see anything wrong with the general approach, any fundamental reason why this can't possibly work, even with tinkering to fix specific flaws? Can anyone think of something that satisfies these axioms but isn't a manifold under some standard definition?

The next step would be to see if I can prove that this definition is equivalent other definitions. I think I can basically do this in the case of one dimension, although I haven't rigorously filled in all the steps. My argument doesn't explicitly make use of M3, which makes me think that there may be some hidden flaw in it (unless M3 somehow is only needed in 2 or more dimensions...?). I could sketch my argument, but it might make more sense to get general comments first on whether I'm on the right track, or reinventing the wheel.

Thanks in advance for any help!

-Ben

A manifold is a Hausdorff, first-countable topological space T such that:

(M1) Self-similarity: Given any two points in T, any sufficiently small open neighborhoods around these points are homeomorphic.

(M2) Density: T is completely metrizable.

(M3) Not fractal: T's Hausdorff dimension and Lebesgue covering dimension are equal.

([EDIT] This set of axioms has problems. #23 is an attempt at fixing them.)

I intend M1 to rule out things like a a manifold with boundary, or a line glued onto a plane, and M2 to rule out things like the rationals.

Does anyone see anything wrong with the general approach, any fundamental reason why this can't possibly work, even with tinkering to fix specific flaws? Can anyone think of something that satisfies these axioms but isn't a manifold under some standard definition?

The next step would be to see if I can prove that this definition is equivalent other definitions. I think I can basically do this in the case of one dimension, although I haven't rigorously filled in all the steps. My argument doesn't explicitly make use of M3, which makes me think that there may be some hidden flaw in it (unless M3 somehow is only needed in 2 or more dimensions...?). I could sketch my argument, but it might make more sense to get general comments first on whether I'm on the right track, or reinventing the wheel.

Thanks in advance for any help!

-Ben

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