
theorem
  for M be non empty MetrSpace,
      S be non empty Subset of M,
      T be non empty Subset of TopSpaceMetr M
    st T = S holds
     (TopSpaceMetr M) | T is countably_compact
     iff S is sequentially_compact
  proof
    let M be non empty MetrSpace,
        S be non empty Subset of M,
        T be non empty Subset of TopSpaceMetr M;
    assume
    A1: T = S;
    hereby
      assume (TopSpaceMetr M) | T is countably_compact; then
      TopSpaceMetr (M|S) is countably_compact by A1,HAUSDORF:16; then
      (M|S) is sequentially_compact by Th11,COMPL_SP:35;
      hence S is sequentially_compact by Th14;
    end;
    assume S is sequentially_compact; then
    (M|S) is sequentially_compact by Th14; then
    TopSpaceMetr (M|S) is countably_compact by Th11,COMPL_SP:35;
    hence (TopSpaceMetr M) |T is countably_compact by A1,HAUSDORF:16;
  end;
