Application of Urysohn's Lemma.

Question

There is one argument of Urysohn's lemma I do not understand.

Assume we have a compact Hausdorff space, so we are in a normal space.

Assume that we have a closed set F, and an open set V, such that $F \subset V$. Then my book states that we can find a continuous function sent to $[0,1]$, such that $f(\{F\})=1$, and supp $f\subset V$.

But my problem is this:

We have two closed sets, F, and $V^c$, these are also disjoint. So Urysohn's lemma states that we can find f, such that $f(\{F\})=1$, and $f(\{V^C\})=0$. Now we have that the set $\{x: f(x) \ne 0\}\subset V$, but in order to get our result, we must have that the closure of this set is in V, and how does this also hold when we take the closure?

Answer 1

It doesn’t: you need an intermediate step. Use the normality of the space to conclude that there is an open set $U$ such that $F\subseteq U\subseteq\operatorname{cl}U\subseteq V$, and then take a Uryson function for $F$ and $X\setminus U$.

Answer 2

Instead of using the function $f$ you get from Urysohn's lemma directly (which indeed might not work), you can use $g(x)=\max(2f(x)-1,0)$.