
theorem
  for M be non empty MetrSpace,
      X be Subset of TopSpaceMetr M,
      p be Element of M holds
    (p in Cl(X) iff for r be Real st 0 < r holds X meets Ball(p,r))
  proof
    let M be non empty MetrSpace,
        X be Subset of TopSpaceMetr M,
        p be Element of M;
    hereby
      assume
      A1: p in Cl(X);
      let r be Real;
      assume
      A2: 0 < r;
      reconsider G = Ball(p,r) as Subset of TopSpaceMetr M;
      dist(p,p) = 0 by METRIC_1:1; then
      G is open & p in G by A2,METRIC_1:11,TOPMETR:14;
      hence X meets Ball(p,r) by A1,PRE_TOPC:def 7;
    end;
    assume
A3: for r be Real st 0 < r holds X meets Ball(p,r);
    now
      let G be Subset of TopSpaceMetr M;
      assume G is open & p in G; then
      consider r be Real such that
      A4: 0 < r & Ball(p,r) c= G by PCOMPS_1:def 4;
      X meets Ball(p,r) by A3,A4;
      hence X meets G by A4,XBOOLE_1:63;
    end;
    hence p in Cl(X) by PRE_TOPC:def 7;
  end;
