let T be non empty TopSpace; :: thesis: for F being Subset-Family of T st F is closed holds
FinMeetCl F is closed
let F be Subset-Family of T; :: thesis: ( F is closed implies FinMeetCl F is closed )
assume A1: F is closed ; :: thesis: FinMeetCl F is closed
now :: thesis: for P being Subset of T st P in FinMeetCl F holds
P is closed
let P be Subset of T; :: thesis: ( P in FinMeetCl F implies P is closed )
assume P in FinMeetCl F ; :: thesis: P is closed
then consider Y being Subset-Family of T such that
A2: Y c= F and
Y is finite and
A3: P = Intersect Y by CANTOR_1:def 3;
A4: ( ( P = the carrier of T & the carrier of T = [#] T ) or P = meet Y ) by A3, SETFAM_1:def 9;
for A being Subset of T st A in Y holds
A is closed by A1, A2;
hence P is closed by A4, PRE_TOPC:14; :: thesis: verum
end;
hence FinMeetCl F is closed ; :: thesis: verum