
theorem Th38:
  for T being non empty TopSpace, x being Point of T, X being Subset of T
  for K being prebasis of T
  st for Z being finite Subset-Family of T st Z c= K & x in Intersect Z
  holds Intersect Z meets X holds x in Cl X
proof
  let T be non empty TopSpace, x be Point of T, X be Subset of T;
  let BB be prebasis of T such that
A1: for Z being finite Subset-Family of T st Z c= BB & x in Intersect Z
  holds Intersect Z meets X;
  reconsider BB9 = FinMeetCl BB as Basis of T by Th23;
  now
    let A be Subset of T;
    assume A in BB9;
    then
A2: ex Y being Subset-Family of T st ( Y c= BB)&( Y is finite)&
    ( A = Intersect Y) by CANTOR_1:def 3;
    assume x in A;
    hence A meets X by A1,A2;
  end;
  hence thesis by Th37;
end;
