
theorem Th16:
  for M be non empty MetrSpace,
      S be Subset of M,
      T be Subset of TopSpaceMetr M
    st T = S holds
    T is compact iff S is sequentially_compact
  proof
    let M be non empty MetrSpace,
        S0 be Subset of M,
        T0 be Subset of TopSpaceMetr M;
    assume
A1: T0 = S0;
    per cases;
    suppose
      T0 = {};
      hence thesis by A1;
    end;
    suppose
  A2: T0 <> {}; then
      reconsider T = T0 as non empty Subset of TopSpaceMetr M;
      reconsider S = T0 as non empty Subset of M by A2;
      hereby
        assume T0 is compact; then
        (TopSpaceMetr M) | T is compact; then
        TopSpaceMetr (M|S) is compact by HAUSDORF:16; then
        (M|S) is sequentially_compact by Th11;
        hence S0 is sequentially_compact by A1,Th14;
      end;
      assume S0 is sequentially_compact; then
      (M|S) is sequentially_compact by A1,Th14; then
      TopSpaceMetr (M|S) is compact by Th11; then
      (TopSpaceMetr M) | T is compact by HAUSDORF:16;
      hence T0 is compact by COMPTS_1:3;
    end;
  end;
