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

proof

assume A1: T is countably_compact ; :: thesis: for F being Subset-Family of st F is centered & F is closed & F is countable holds
meet F <> {}
let F be Subset-Family of ; :: thesis: ( F is centered & F is closed & F is countable implies meet F <> {} )
assume that
A2: F is centered and
A3: F is closed and
A4: F is countable ; :: thesis: meet F <> {}
assume A5: meet F = {} ; :: thesis: contradiction
F <> {} by A2, FINSET_1:def 3;
then union (COMPLEMENT F) = (meet F) ` by TOPS_2:12
.= [#] T by A5 ;
then A6: COMPLEMENT F is Cover of by SETFAM_1:60;
A7: COMPLEMENT F is countable by A4, TOPGEN_4:1;
COMPLEMENT F is open by A3, TOPS_2:16;
then consider G' being Subset-Family of such that
A8: G' c= COMPLEMENT F and
A9: G' is Cover of and
A10: G' is finite by A1, A6, A7, Def9;
A11: COMPLEMENT G' is finite by A10, TOPS_2:13;
A12: 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 A13: x in COMPLEMENT G' ; :: thesis: x in F
reconsider x' = x as Subset of by A13;
x' ` in G' by A13, SETFAM_1:def 8;
then (x' ` ) ` in F by A8, SETFAM_1:def 8;
hence x in F ; :: thesis: verum

end;

G' <> {} by A9, TOPS_2:5;
then A14: COMPLEMENT G' <> {} by TOPS_2:10;
meet (COMPLEMENT G') = (union G') ` by A9, TOPS_2:5, TOPS_2:11
.= ([#] T) \ ([#] T) by A9, SETFAM_1:60
.= {} by XBOOLE_1:37 ;
hence contradiction by A2, A12, A14, A11, FINSET_1:def 3; :: thesis: verum

end;

assume A15: for F being Subset-Family of st F is centered & F is closed & F is countable holds
meet F <> {} ; :: thesis: T is countably_compact
let F be Subset-Family of ; :: according to COMPL_SP:def 9 :: thesis: ( F is Cover of & F is open & F is countable implies ex G being Subset-Family of st
( G c= F & G is Cover of & G is finite ) )
assume that
A16: F is Cover of and
A17: F is open and
A18: F is countable ; :: thesis: ex G being Subset-Family of st
( G c= F & G is Cover of & G is finite )
A19: COMPLEMENT F is countable by A18, TOPGEN_4:1;
F <> {} by A16, TOPS_2:5;
then A20: COMPLEMENT F <> {} by TOPS_2:10;
A21: COMPLEMENT F is closed by A17, TOPS_2:17;
meet (COMPLEMENT F) = (union F) ` by A16, TOPS_2:5, TOPS_2:11
.= ([#] T) \ ([#] T) by A16, SETFAM_1:60
.= {} by XBOOLE_1:37 ;
then not COMPLEMENT F is centered by A15, A19, A21;
then consider G' being set such that
A22: G' <> {} and
A23: G' c= COMPLEMENT F and
A24: G' is finite and
A25: meet G' = {} by A20, FINSET_1:def 3;
reconsider G' = G' as Subset-Family of by A23, 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 x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in F' or x in F )
assume A26: x in F' ; :: thesis: x in F
reconsider x' = x as Subset of by A26;
x' ` in G' by A26, SETFAM_1:def 8;
then (x' ` ) ` in F by A23, SETFAM_1:def 8;
hence x in F ; :: thesis: verum

end;

union F' = (meet G') ` by A22, TOPS_2:12
.= [#] T by A25 ;
hence ( F' is Cover of & F' is finite ) by A24, SETFAM_1:60, TOPS_2:13; :: thesis: verum