reserve n for Nat;

theorem
  for A being Subset of TOP-REAL n, p being Point of TOP-REAL n, p9
  being Point of Euclid n st p = p9 holds p in Cl A iff for r being Real
  st r > 0 holds Ball (p9, r) meets A
proof
  let A be Subset of TOP-REAL n, p be Point of TOP-REAL n, p9 be Point of
  Euclid n;
  assume
A1: p = p9;
  hereby
    assume
A2: p in Cl A;
    let r be Real;
    reconsider B = Ball (p9, r) as Subset of TOP-REAL n by TOPREAL3:8;
    assume r > 0;
    then B is a_neighborhood of p by A1,GOBOARD6:2;
    hence Ball (p9, r) meets A by A2,CONNSP_2:27;
  end;
  assume
A3: for r being Real st r > 0 holds Ball (p9, r) meets A;
  for G being a_neighborhood of p holds G meets A
  proof
    let G be a_neighborhood of p;
    p in Int G by CONNSP_2:def 1;
    then ex r being Real st r > 0 & Ball (p9, r) c= G by A1,GOBOARD6:5;
    hence thesis by A3,XBOOLE_1:63;
  end;
  hence thesis by CONNSP_2:27;
end;
