Question

Find necessary and sufficient conditions for A to be the complement of a compact set.

Answer 1

A subset A of topological space X is compact if for every open cover of A there exists a finite sub cover of A.

Also, a set is said to be compact iff it is closed and bounded.

And a complement of a set A, denoted A', is the set of all elements in the given universal set U that are not in A.

Since complement of a closed set is open, so complement of a compact set must be open.

So a necessary condition is the set should be open.

Now consider that the complement of a set is bounded if the set is bounded.

Thus, putting it all together, the necessary and sufficient condition for a set be complement of a compact set, is it must be open and bounded.