let T be non empty TopSpace; :: thesis: ( T is compact iff for F being Subset-Family of st F is centered & F is closed holds
meet F <> {} )
thus ( T is compact implies for F being Subset-Family of st F is centered & F is closed holds
meet F <> {} ) :: thesis: ( ( for F being Subset-Family of st F is centered & F is closed holds
meet F <> {} ) implies T is compact )

proof

assume A1: T is compact ; :: thesis: for F being Subset-Family of st F is centered & F is closed holds
meet F <> {}
let F be Subset-Family of ; :: thesis: ( F is centered & F is closed implies meet F <> {} )
assume that
A2: F is centered and
A3: F is closed ; :: thesis: meet F <> {}
reconsider C = COMPLEMENT F as Subset-Family of ;
assume A4: meet F = {} ; :: thesis: contradiction
F <> {} by A2, FINSET_1:def 3;
then union (COMPLEMENT F) = (meet F) ` by TOPS_2:12
.= [#] T by A4 ;
then A5: COMPLEMENT F is Cover of by SETFAM_1:def 12;
COMPLEMENT F is open by A3, TOPS_2:16;
then consider G' being Subset-Family of such that
A6: G' c= C and
A7: G' is Cover of and
A8: G' is finite by A1, A5, Def3;
set F' = COMPLEMENT G';
A9: COMPLEMENT G' is finite by A8, TOPS_2:13;
A10: COMPLEMENT G' c= F

proof

let X be set ; :: according to TARSKI:def 3 :: thesis: ( not X in COMPLEMENT G' or X in F )
assume A11: X in COMPLEMENT G' ; :: thesis: X in F
then reconsider X1 = X as Subset of ;
X1 ` in G' by A11, SETFAM_1:def 8;
then (X1 ` ) ` in F by A6, SETFAM_1:def 8;
hence X in F ; :: thesis: verum

end;

G' <> {} by A7, TOPS_2:5;
then A12: COMPLEMENT G' <> {} by TOPS_2:10;
meet (COMPLEMENT G') = (union G') ` by A7, TOPS_2:5, TOPS_2:11
.= ([#] T) \ ([#] T) by A7, SETFAM_1:60
.= {} by XBOOLE_1:37 ;
hence contradiction by A2, A10, A9, A12, FINSET_1:def 3; :: thesis: verum

end;

assume A13: for F being Subset-Family of st F is centered & F is closed holds
meet F <> {} ; :: thesis: T is compact
thus T is compact :: thesis: verum

proof

let F be Subset-Family of ; :: according to COMPTS_1:def 3 :: thesis: ( F is Cover of & F is open implies ex G being Subset-Family of st
( G c= F & G is Cover of & G is finite ) )
assume that
A14: F is Cover of and
A15: F is open ; :: thesis: ex G being Subset-Family of st
( G c= F & G is Cover of & G is finite )
reconsider G = COMPLEMENT F as Subset-Family of ;
A16: G is closed by A15, TOPS_2:17;
F <> {} by A14, TOPS_2:5;
then A17: G <> {} by TOPS_2:10;
meet G = (union F) ` by A14, TOPS_2:5, TOPS_2:11
.= ([#] T) \ ([#] T) by A14, SETFAM_1:60
.= {} by XBOOLE_1:37 ;
then not G is centered by A13, A16;
then consider G' being set such that
A18: G' <> {} and
A19: G' c= G and
A20: G' is finite and
A21: meet G' = {} by A17, FINSET_1:def 3;
reconsider G' = G' as Subset-Family of by A19, XBOOLE_1:1;
take F' = COMPLEMENT G'; :: thesis: ( F' c= F & F' is Cover of & F' is finite )
thus F' c= F :: thesis: ( F' is Cover of & F' is finite )

proof

let A be set ; :: according to TARSKI:def 3 :: thesis: ( not A in F' or A in F )
assume A22: A in F' ; :: thesis: A in F
then reconsider A1 = A as Subset of ;
A1 ` in G' by A22, SETFAM_1:def 8;
then (A1 ` ) ` in F by A19, SETFAM_1:def 8;
hence A in F ; :: thesis: verum

end;

union F' = (meet G') ` by A18, TOPS_2:12
.= [#] T by A21 ;
hence F' is Cover of by SETFAM_1:def 12; :: thesis: F' is finite
thus F' is finite by A20, TOPS_2:13; :: thesis: verum

end;