let T be non empty TopSpace; :: thesis: for A, B being Subset of T st A is F_sigma & B is F_sigma holds
A /\ B is F_sigma
let A, B be Subset of T; :: thesis: ( A is F_sigma & B is F_sigma implies A /\ B is F_sigma )
assume that
A1: A is F_sigma and
A2: B is F_sigma ; :: thesis: A /\ B is F_sigma
consider F being countable closed Subset-Family of T such that
A3: A = union F by A1, Def6;
consider G being countable closed Subset-Family of T such that
A4: B = union G by A2, Def6;
reconsider H = INTERSECTION F,G as Subset-Family of T ;
A5: H is closed by Th21;
( card H c= card [:F,G:] & [:F,G:] is countable ) by Th25, CARD_4:55;
then A6: H is countable by WAYBEL12:2;
A /\ B = union H by A3, A4, SETFAM_1:39;
hence A /\ B is F_sigma by A5, A6, Def6; :: thesis: verum