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Homework Statement
Let X be a normal space. There exists a continuous function h : X --> [0, 1] such that h(A) = 0, h(B) = 1 and h(X\(AUB))[tex]\in[/tex]<0, 1> iff A and B are disjoint closed Gδ sets in X.
The Attempt at a Solution
I'll only display my attempt for one direction, since the other one is rather easy.
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Let A and B be disjoint closed Gδ sets in X. Since their union is a Gδ set, too, there exists a function f : X --> [0, 1] such that f(AUB) = 0 and f(X\(AUB)) > 0. Next, for every positive integer m, apply the Urysohn lemma to the sets X\Bm (since B is Gδ, the sets Bm are open members of the family whose countable intersection equals B) and B in order to obtain a continuous function gm : X --> [0, 1] such that gm(X\Bm) = 0 and gm(B) = 1.
Now, for every positive integer, define the function hm(x) = 1/2 f(x) + gm(x). I'm not really sure about this, but the function we're looking for could be h(x) = lim hm(x), as n --> ∞. If x is in A, for m large enough (since Bm could actually intersect A) hm(A) = 0, hm(B) = 1, and any x not in A or B should eventually converge to 1/2f(x), which is in <0, 1>.
I hope this works, thanks in advance.