Topology: Munkres - Urysohn lemma

[Fisicks]

Hi, the problem I am referencing is section 33 problem 4.

Let X be normal. There exists a continuous function f: X -> [0,1] such that fx=0 for x in A and fx >0 for x not in A, if and only if A is a closed G(delta) set in X.

My question is about the <= direction.

So let B be the collection of open sets whose intersection is A and index them with the natural numbers. Let U_n be an element of B. For each U_n, define a function f_n. To define f_n follow the proof of Urysohn lemma using A=A, B=X-U_n.

Define fx= sup{f_n(x)} for all n.

Clearly fx=0 iff x is in A. My problem is with showing continuity. Part of me thinks that if x is an element of X and (a,b) is a basic open set of fx, then there exists an open set U such fU is contained in (a,b) since each f_n is continuous.

 

[fresh_42]

I think we need a reference here as Urysohn doesn't make a statement about the values of $f$ outside $A$ and $B$. Also "G(delta)" needs to be explained.