
theorem Th24:
  for T being non empty TopSpace st T is T_2 for S being sequence
  of T st S is convergent holds ex x being Point of T st Lim S = {x}
proof
  let T be non empty TopSpace;
  assume
A1: T is T_2;
  let S be sequence of T;
  assume S is convergent;
  then consider x being Point of T such that
A2: S is_convergent_to x;
  take x;
A3: x in Lim S by A2,FRECHET:def 5;
  thus Lim S c= {x}
  proof
    let y be object;
    assume
A4: y in Lim S;
    then reconsider y9=y as Point of T;
    y9=x by A1,A3,A4,Th6;
    hence thesis by TARSKI:def 1;
  end;
  let y be object;
  assume y in {x};
  hence thesis by A3,TARSKI:def 1;
end;
