If A and B are subsets of a metric space (X, d), the distance between them, d(A,B), is defined as d(A, B) = infimum {d(a, b) | a is in A, b is in B}. Show that:
a) If the closures of A and B intersect, then d(A,B) = 0. Is the converse true?
b) If A is compact, B is closed and A and B are disjoint, then d(A,B) > 0. If, instead of assuming A is compact, we just assume it's closed, then is it still true that d(A,B) > 0?
Thank you
2007-11-30
01:56:24
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4 answers
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asked by
Edson
1
in
Mathematics