Show that any closed subset of a compact topological space i

Show that any closed subset of a compact topological space is compact.

Solution

If STST and TT is compact and SS is closed then SS is compact.

Proof: Let UU be an open cover of SS. Every open set in UU is of the form USUS for some open set UU(open in TT). Let V={UTU is open, and UU:US=U}V={UTU is open, and UU:US=U}. Then VV is an open cover of SS as well, since SS is closed we have that TSTS is open so V{TS}V{TS} is an open cover of TT.

By compactness of TT we have a finite subcover, from which we can produce a finite subcover of UU.

Please note that, we have shown that every open cover of SS has a finite subcover, and therefore SS is compact. We have used the fact that SS is closed to make sure that TSTS is open. If SS is not closed we cannot use this to produce an open cover of TT and we cannot continue and find an open subcover for UU.