hw3additional-555-2011 - A a Prove d x,A = 0 if and only if...

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Homework 3, Additional Problems. (1) Let ( X,d ) be a metric space. a. Let E i X ( i ∈ { 1 , ··· ,n } ) be a ﬁnite collection of subsets of X. Prove that n i =1 E i = n i =1 E i . b. Let E i ( i I ) now be an arbitrary collection of subsets of X . Prove that i I E i ⊂ ∩ i I E i and give an example that in general (even for two sets) that the inclusion is proper. (2) Let ( X,d ) be a metric space and let A X be a non-empty subset. Deﬁne d ( x,A ) = inf { d ( x,y ) : y
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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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