let TS be TopSpace; :: thesis: for PS, QS being Subset of TS st TS is T_2 & PS is compact & QS is compact holds
PS /\ QS is compact
let PS, QS be Subset of TS; :: thesis: ( TS is T_2 & PS is compact & QS is compact implies PS /\ QS is compact )
assume that
A1: TS is T_2 and
A2: PS is compact and
A3: QS is compact ; :: thesis: PS /\ QS is compact
A4: QS is closed by A1, A3, Th7;
PS is closed by A1, A2, Th7;
hence PS /\ QS is compact by A2, A4, Th9, XBOOLE_1:17; :: thesis: verum