reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem Th19:
  F = { A } implies (A is open iff F is open)
proof
  assume
A1: F = { A };
  hereby
    assume A is open;
    then for A being Subset of T st A in F holds A is open by A1,TARSKI:def 1;
    hence F is open by TOPS_2:def 1;
  end;
  assume
A2: F is open;
  A in F by A1,TARSKI:def 1;
  hence thesis by A2,TOPS_2:def 1;
end;
