let Y be non empty TopSpace; :: thesis: for x being Point of Y holds MaxADSet x c= meet { F where F is Subset of Y : ( F is closed & x in F ) }
let x be Point of Y; :: thesis: MaxADSet x c= meet { F where F is Subset of Y : ( F is closed & x in F ) }
set G = { F where F is Subset of Y : ( F is closed & x in F ) } ;
[#] Y in { F where F is Subset of Y : ( F is closed & x in F ) } ;
then A1: { F where F is Subset of Y : ( F is closed & x in F ) } <> {} ;
{ F where F is Subset of Y : ( F is closed & x in 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 & x in F ) } or C in bool the carrier of Y )
assume C in { F where F is Subset of Y : ( F is closed & x in F ) } ; :: thesis: C in bool the carrier of Y
then ex P being Subset of Y st
( C = P & P is closed & x in 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 & x in F ) } as Subset-Family of Y ;
now :: thesis: for C being set st C in G holds