Suppose that X satisfies the Bolzano Weierstrass Property and that A and B are compact subsets of X Prove that if A B and if dist (A, B) inf p(x, y) x A and y B , then dist (A, B) 0 Show that even in the space R2, there exist subsets A and B which are closed and satisfy A B , but dist(A, B) 0 ...

a Since both sets are nonempty and px y is bounded ...

Suppose that X satisfies the Bolzano-Weierstrass Property and that A and B are compact subsets of X.

Question:

Suppose that X satisfies the Bolzano-Weierstrass Property and that A and B are compact subsets of X. Prove that if A ∩ B = θ and if
dist (A, B) := inf{p(x, y) : x ∈ A and y ∈ B},
then dist (A, B) > 0. Show that even in the space R2, there exist subsets A and B which are closed and satisfy A ∩ B = θ, but dist(A, B) = 0.