What is meant by a normal topological space? State and prove Urysohn's lemma.

Let $X$ be a normal topological space and let $S \subseteq X$ be closed. Show that there is a continuous function $f: X \rightarrow[0,1]$ with $f^{-1}(0)=S$ if, and only if, $S$ is a countable intersection of open sets.

[Hint. If $S=\bigcap_{n=1}^{\infty} U_{n}$ then consider $\sum_{n=1}^{\infty} 2^{-n} f_{n}$, where the functions $f_{n}: X \rightarrow[0,1]$ are supplied by an appropriate application of Urysohn's lemma.]