
theorem Th6:
  for M being non empty MetrSpace, p being Point of M, A being
  Subset of TopSpaceMetr M holds p in Cl A iff for r being Real st r > 0
  ex q being Point of M st q in A & dist (p, q) < r
proof
  let M be non empty MetrSpace, p be Point of M, A be Subset of TopSpaceMetr M;
  hereby
    assume
A1: p in Cl A;
    let r be Real;
    assume r > 0;
    then Ball (p, r) meets A by A1,GOBOARD6:92;
    then consider x being object such that
A2: x in Ball (p, r) and
A3: x in A by XBOOLE_0:3;
    reconsider q = x as Point of M by A2;
    take q;
    thus q in A by A3;
    thus dist (p, q) < r by A2,METRIC_1:11;
  end;
  assume
A4: for r being Real st r > 0 ex q being Point of M st q in A &
  dist (p, q) < r;
  for r being Real st r > 0 holds Ball (p, r) meets A
  proof
    let r be Real;
    assume r > 0;
    then consider q being Point of M such that
A5: q in A and
A6: dist (p, q) < r by A4;
    q in Ball (p, r) by A6,METRIC_1:11;
    hence thesis by A5,XBOOLE_0:3;
  end;
  hence thesis by GOBOARD6:92;
end;
