N m is called an immersion if the rank of f is 11, the dimension of the domain M, at all points of M. : 11, and it is clear from Theorem 10(2) that an immersion is locally one-one (so it is a topological immersion, as defined in Chapter I).

B) Given a homeomorphism � � x � x . . and a � � which = � � x (� x � x . . ) by goes to co at co, define Show that is closed. (c) There are at most C non-homeomorphic connected manifolds (where c = 2No is the cardinality of �). j(M) J j: M J: M x J(x ) n g: M j(x) (g(x),J(x». 24. (a) It is possible for �2 - A and �2 - Bto be homeomorphic even though A and B are non-homeomorphic closed subsets. (b) If A c �2 is closed and totally disconnected (the only components of A are points), then e(�2 - A) is homeomorphic to A.

### A Comprehensive Introduction To Differential Geometry by Michael Spivak

by Jeff

4.4