Let

be a metric space. For arbitrary

and

put

It can be proved, if

is compact space, then for every tho closed and disjoint subsets

and

is

Prove that in conected metric spaces holds converse, naimly, if for every two closed and disjoint subsets... this distance is positive, than

is compact.

Condition conected on metric space can't be omitted.

Hint: My proff is relativly long and is based on contrary asumptation that

isn't compact, and then using knowing equivalent of compacity in terms of sequences, conscruct efectivly tho closed and disjoint subsets

and

for which is

. But there are some other equivalents of compacity, maybe someone find another solution. I have the appropriate counterexemple of totaly disconected metric space in which is

for suitable subsets, but space isn't compact.