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Jamin2112
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(a) If A and B are disjoint closed sets in some ...
(a) If A and B are disjoint closed sets in some metric space X, prove that they are separated.
(b) Prove the same for disjoint open sets.
(c) Fix p in X, ∂ >0, define A to be the set of all q in Z for which d(p,q) < ∂, define B similarly, with > in place of <. Prove that A and B are separated.
Triangle inequality, definitions of open and closed sets
Give me some hints to finishing part (c).
Homework Statement
(a) If A and B are disjoint closed sets in some metric space X, prove that they are separated.
(b) Prove the same for disjoint open sets.
(c) Fix p in X, ∂ >0, define A to be the set of all q in Z for which d(p,q) < ∂, define B similarly, with > in place of <. Prove that A and B are separated.
Homework Equations
Triangle inequality, definitions of open and closed sets
The Attempt at a Solution
Give me some hints to finishing part (c).