The usual definition of an n-dimensional topological manifold M is a topological space which is 'locally Euclidean', by which we mean that:(adsbygoogle = window.adsbygoogle || []).push({});

(1) every point in M is contained in an open set which is homeomorphic to ##\mathbb{R}^n##.

(2) M is second countable.

(3) M is an Hausdorff space.

Could someone explain what the idea is behind (2) and (3)? Normally I would think that locally Euclidean would refer to R^n with it's usual topology (the one induced by the Euclidean metric), but the definition above seems to be more general. It might even not be metrizable. According to http://en.wikipedia.org/wiki/Metrization_theorem one does also require the space to be 'regular'. Is perhaps regularity implied by (1)?

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# Idea behind topological manifold definition.

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