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## Homework Statement

Let (X, p) be a metric space, and let A and B be nonempty, closed, disjoint subsets of X.

define d(x,A) = inf{p(x, a)|a in A}

h(x) = d(x, A)/[d(x, A) + d(x, B)]

defines a continuous function h: X -> [0,1]. h(x) = 0 iff x is in A, and h(x) = 1 iff x is in B. Infer that there exist open sets U and V of X such that [tex]A \subset U[/tex] and [tex]B \subset [/tex]V with [tex]U \cap V = \emptyset [/tex]

## Homework Equations

## The Attempt at a Solution

I have showed everything except the last part about disjoint open sets. I don't think I can just say that since A and B are disjoint, there is an r>0 such that [tex]B_r(A) \cap B_r(B) = \emptyset[/tex]. I'm actually having a hard time seeing how the iff statements are necessary. Any hints would be helpful.