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