let Y be non empty TopSpace; :: thesis: for A being Subset of Y holds MaxADSet A c= meet { F where F is Subset of Y : ( F is closed & A c= F ) }
let A be Subset of Y; :: thesis: MaxADSet A c= meet { F where F is Subset of Y : ( F is closed & A c= F ) }
set G = { F where F is Subset of Y : ( F is closed & A c= F ) } ;
{ F where F is Subset of Y : ( F is closed & A c= F ) } c= bool the carrier of Y
proof
let C be object ; :: according to TARSKI:def 3 :: thesis: ( not C in { F where F is Subset of Y : ( F is closed & A c= F ) } or C in bool the carrier of Y )
assume C in { F where F is Subset of Y : ( F is closed & A c= F ) } ; :: thesis: C in bool the carrier of Y
then ex P being Subset of Y st
( C = P & P is closed & A c= P ) ;
hence C in bool the carrier of Y ; :: thesis: verum
end;
then reconsider G = { F where F is Subset of Y : ( F is closed & A c= F ) } as Subset-Family of Y ;
A1: now :: thesis: for C being set st C in G holds
MaxADSet A c= C
let C be set ; :: thesis: ( C in G implies MaxADSet A c= C )
assume C in G ; :: thesis:
then ex F being Subset of Y st
( F = C & F is closed & A c= F ) ;
hence MaxADSet A c= C by Th40; :: thesis: verum
end;
[#] Y in G ;
then G <> {} ;
hence MaxADSet A c= meet { F where F is Subset of Y : ( F is closed & A c= F ) } by ; :: thesis: verum