Topological properties of manifolds

We now proceed to the constructions that are directly related to what we shall later use

Theorem A smooth manifold has a compact exhaustion and is paracompact

A compact exhaustion is an increasing countable collection of compact sets K_1\subset K_2\subset\dots such that M=\bigcup K_i and K_i\subset\mathrm{int}K_{i+1} for all i.

The fundamental lemma we need is a smooth version of Urysohn’s lemma

Theorem (Smooth Urysohn Lemma) If M is a smooth manifold and A,B\subset M are disjoint closed sets, then there exist a smooth function f:M\to [0,1] such that A=f^{-1}(0 and B=f^{-1}(1).

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