# 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 $B=f^{-1}(1)$.