let X be non empty TopSpace; :: thesis: ( X is extremally_disconnected iff for A, B being Subset of X st A is closed & B is closed & A \/ B = the carrier of X holds
(Int A) \/ (Int B) = the carrier of X )
thus ( X is extremally_disconnected implies for A, B being Subset of X st A is closed & B is closed & A \/ B = the carrier of X holds
(Int A) \/ (Int B) = the carrier of X ) :: thesis: ( ( for A, B being Subset of X st A is closed & B is closed & A \/ B = the carrier of X holds
(Int A) \/ (Int B) = the carrier of X ) implies X is extremally_disconnected )
proof
assume A1: X is extremally_disconnected ; :: thesis: for A, B being Subset of X st A is closed & B is closed & A \/ B = the carrier of X holds
(Int A) \/ (Int B) = the carrier of X
let A, B be Subset of X; :: thesis: ( A is closed & B is closed & A \/ B = the carrier of X implies (Int A) \/ (Int B) = the carrier of X )
assume that
A2: A is closed and
A3: B is closed ; :: thesis: ( not A \/ B = the carrier of X or (Int A) \/ (Int B) = the carrier of X )
assume A \/ B = the carrier of X ; :: thesis: (Int A) \/ (Int B) = the carrier of X
then (A \/ B) ` = {} X by XBOOLE_1:37;
then (A `) /\ (B `) = {} X by XBOOLE_1:53;
then A ` misses B ` ;
then Cl (A `) misses Cl (B `) by A1, A2, A3, Th28;
then (Cl (A `)) /\ (Cl (B `)) = {} X ;
then ((Cl (A `)) /\ (Cl (B `))) ` = [#] X ;
then ((Cl (A `)) `) \/ ((Cl (B `)) `) = [#] X by XBOOLE_1:54;
then ((Cl (A `)) `) \/ (Int B) = [#] X by TOPS_1:def 1;
hence (Int A) \/ (Int B) = the carrier of X by TOPS_1:def 1; :: thesis: verum
end;
assume A4: for A, B being Subset of X st A is closed & B is closed & A \/ B = the carrier of X holds
(Int A) \/ (Int B) = the carrier of X ; :: thesis: X is extremally_disconnected
now :: thesis: for A, B being Subset of X st A is open & B is open & A misses B holds
Cl A misses Cl B
let A, B be Subset of X; :: thesis: ( A is open & B is open & A misses B implies Cl A misses Cl B )
assume that
A5: A is open and
A6: B is open ; :: thesis: ( A misses B implies Cl A misses Cl B )
assume A misses B ; :: thesis: Cl A misses Cl B
then A /\ B = {} X ;
then (A /\ B) ` = [#] X ;
then (A `) \/ (B `) = [#] X by XBOOLE_1:54;
then (Int (A `)) \/ (Int (B `)) = the carrier of X by A4, A5, A6;
then ((Int (A `)) \/ (Int (B `))) ` = {} X by XBOOLE_1:37;
then ((Int (A `)) `) /\ ((Int (B `)) `) = {} X by XBOOLE_1:53;
then (Cl A) /\ ((Int (B `)) `) = {} X by Th1;
then Cl A misses (Int (B `)) ` ;
hence Cl A misses Cl B by Th1; :: thesis: verum
end;
hence X is extremally_disconnected by Th28; :: thesis: verum