1. (a) If A is a compact subset of a metric space (X, d ) and B is a closed subset of A, prove that B is also compact. (b) Prove that the intersection of any collection of compact sets in a metric...

1. (a) If A is a compact subset of a metric space (X, d ) and B is a closed subset of A, prove that B is also compact.

(b) Prove that the intersection of any collection of compact sets in a metric space is compact.

2. Let A be a subset of a metric space (X, d ). Prove the following:

(a) If A is open, then int (bd A) = ∅.

(b) If A is closed, then int (bd A) = ∅.

(c) Find an example of a metric space (X, d ) and a subset A such that int (bd A) = X.