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I just started studying smooth manifolds. The definition of a topological manifold requires a topological space to be locally Euclidean: homeomorphic to $\mathbb{R}^n$.

I know some examples, like how a 2-sphere is locally homeomorphic to $\mathbb{R}^2$. In this case we have an intuitive notion of why $n=2$.

Question: for a general topological space, how do we know what $n$ to choose?

Jean Valjean
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  • Perhaps "Lebesgue covering dimension" is a place to start? – Eric Towers Jan 27 '14 at 03:14
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    @Eric Almost certainly not, since general topology is much messier than manifold topology. If you have access to a description of arbitrary open sets you can surely just check whether there are sufficiently small Euclidean sets, and of what dimension. – Ryan Reich Jan 27 '14 at 03:41
  • I think you can first to look what does its open set look like? – gaoxinge Jan 27 '14 at 05:33

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Usually the assumption will state that a manifold is $n$-dimensional. In other cases, it will state that the manifold is given by level sets, or by gluing together other manifolds, and should still usually be clear which $n$ to choose. You mention the sphere as a clear example. Another example that is easy is the projective space, or the graph of a continuous function.

user901823
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