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Unformatted text preview: l topology. 3. Show that Rn with the usual topology is separable and Hausdorff. 13 Coordinate systems and manifolds Suppose M is a topological space, U is an open subset of M , and : U Rn. Suppose further that (U ) is an open subset of Rn, and that is a homeomorphism between U and (U ). We call a local coordinate system of dimension n on U . For each point m U , we then have that (m) = (1(m), . . . , n(m)), the coordinates of m with respect to . Now suppose that we have another open subset V of M , and is a local coordinate system on V . We say that and are C inf ty compatible if the composite functions -1 and -1 are C functions on (U ) (V ). Remember that
14 a function on Rn is C if it is continuous, and all its partial derivatives are also continuous. A topological manifold of dimension n is a separable Hausdorff space M such that every point in M is in the domain of a local coordinate system of dimension n. These spaces are sometimes called locally Euclidean spaces. A C differentiable structure on...
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