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 such that and for all .
The fundamental lemma we need is a smooth version of Urysohn’s lemma
Theorem (Smooth Urysohn Lemma) If is a smooth manifold and are disjoint closed sets, then there exist a smooth function such that and .