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