LectureNotes3G

LectureNotes3G - Math 598 Geometry and Topology II Spring...

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Math 598 Jan 21, 2005 1 Geometry and Topology II Spring 2005, PSU Lecture Notes 3 1.8 Immersions and Embeddings Let X and Y be topological spaces and f : X Y be a continuous map. We say that f is an immersion provided that it is locally one-to-one, and f is an embedding provided that f is a homeomorphism between X and f ( X ) with respect to the subspace topology. Exercise 1. Show that if X is compact, Y is Hausdorf and f : X Y is a one-to-one continuous map, then f is an embedding. Demonstrate, by means of a counterexample, that the previous sentence may not be true if X is not compact. Exercise 2. Show that if f : M N is an immersion between manifolds of the same dimension, then f is open. In particular, if M is compact, and N is connected but not compact, then there exists no immersion f : M N . ( Hint: Use the invariance domain: if two subsets of R n are homeomorphic and one of them is open then the other is open as well. ) Exercise 3. Let C M be a compact set, and
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LectureNotes3G - Math 598 Geometry and Topology II Spring...

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