
theorem Th4:
  for M be non empty MetrSpace,
      X be Subset of TopSpaceMetr M holds
      X is closed
    iff
      for S be sequence of M
        st (for n be Nat holds S.n in X)
         & S is convergent
      holds lim S in X
  proof
    let M be non empty MetrSpace,
        X be Subset of TopSpaceMetr M;
    thus X is closed implies
         for S be sequence of M
           st (for n be Nat holds S . n in X) & S is convergent
         holds lim S in X by TOPMETR3:1;
    assume
    A1: for S be sequence of M
          st (for n be Nat holds S.n in X)
           & S is convergent holds lim S in X;
    for x be object st x in Cl X holds x in X
    proof
      let x be object;
      assume x in Cl X; then
      consider S be sequence of M such that
      A2: (for n be Nat holds S.n in X) & S is convergent & lim S = x by Th3;
      thus x in X by A1,A2;
    end; then
    Cl X = X by PRE_TOPC:18,TARSKI:def 3;
    hence X is closed by PRE_TOPC:22;
  end;
