For any connected topological manifold, it is true that for any two points on the manifold, there exists a single local chart that both of two points lie in it.
How can I prove it?
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Hint: Connected manifolds are path connected. – user135988 Sep 21 '15 at 11:44
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How does use it? – Bimanifold Sep 21 '15 at 11:45
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I think what you are looking for can be found here: http://math.stackexchange.com/questions/35780/given-two-points-in-a-manifold-can-one-always-find-a-topological-ball-that-cont – user135988 Sep 21 '15 at 11:47
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2@user135988: I think the difficulty lies in the fact that the manifold is not assumed to have any differentiable structure... – frog Sep 21 '15 at 11:59
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@frog: Everything in that answer works fine, you just need to use locally flat paths as opposed to smooth ones. – Sep 21 '15 at 15:21
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@Mike: What do you mean by locally flat? – frog Sep 22 '15 at 09:16
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@frog This. – Sep 22 '15 at 14:28
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Ok, I see now. Thanks for this input! – frog Sep 23 '15 at 13:16