Let S and T be subsets of . Find a counterexample for each of the following. (a) If P is the set of all isolated points of S, then P is a closed set. (b) Every open set contains at least two points....

Let S and T be subsets of
. Find a counterexample for each of the following.

(a) If P is the set of all isolated points of S, then P is a closed set.

(b) Every open set contains at least two points.

(c) If S is closed, then cl (int S ) = S.

(d) If S is open, then int (cl S ) = S.

(e) bd (cl S ) = bd S

(f ) bd (bd S ) = bd S

(g) bd (S ∪ T ) = (bd S ) ∪ (bd T )

(h) bd (S ∩ T ) = (bd S ) ∩ (bd T )