Show that the inherited subspace topology is the same

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online