Unformatted text preview: A } . a. Prove d ( x,A ) = 0 if and only if x ∈ A . b. Show that  d ( x,A )d ( y,A )  ≤ d ( x,y ) , for all x,y ∈ X . c. Let now A,B ⊂ X be disjoint closed subsets. Prove there exists a continuous f : X → [0 , 1] such that f ( x ) = 0 for all x ∈ A and f ( x ) = 1 for all x ∈ B . (Hint: Show that f ( x ) = d ( x,A ) d ( x,A )+ d ( x,B ) works.) 1...
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 Spring '07
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 Addition, Sets, Empty set, Topological space, Basic concepts in set theory, Vacuous truth, Ei ii Ei

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