
theorem Th37:
  for T being non empty TopSpace, x being Point of T, X being Subset of T
  for K being Basis of T
  st for A being Subset of T st A in K & x in A holds A 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 Basis of T such that
A1: for A being Subset of T st A in BB & x in A holds A meets X;
  now
    let G be a_neighborhood of x;
    A2: Int
 G = union {A where A is Subset of T: A in BB & A c= G} by YELLOW_8:11;
    x in Int G by CONNSP_2:def 1;
    then consider Z being set such that
A3: x in Z and
A4: Z in {A where A is Subset of T: A in BB & A c= G} by A2,TARSKI:def 4;
    ex A being Subset of T st ( Z = A)&( A in BB)&( A c= G) by A4;
    hence G meets X by A1,A3,XBOOLE_1:63;
  end;
  hence thesis by CONNSP_2:27;
end;
