reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th24:
  for V being Subset of TOP-REAL n holds
  V in the topology of TOP-REAL n iff
  for p st p in V ex r st r > 0 & Ball(p,r) c= V
proof
  let V be Subset of TOP-REAL n;
  reconsider n1=n as Element of NAT by ORDINAL1:def 12;
  set T = TOP-REAL n;
A1: the TopStruct of T = TopSpaceMetr Euclid n by EUCLID:def 8;
A2: TopSpaceMetr Euclid n = TopStruct (#the carrier of Euclid n,
  Family_open_set Euclid n#) by PCOMPS_1:def 5;
  reconsider V1 = V as Subset of Euclid n by EUCLID:67;
  hereby
    assume
A3: V in the topology of T;
    let p;
    assume
A4: p in V;
    reconsider x = p as Element of Euclid n by EUCLID:67;
    consider r be Real such that
A5: r > 0 & Ball(x,r) c= V1 by A3,A4,A1,A2,PCOMPS_1:def 4;
    reconsider r as Real;
    take r;
    thus r > 0 by A5;
    reconsider x1 = x as Point of Euclid n1;
    reconsider p1 = p as Point of TOP-REAL n1;
    thus Ball(p,r) c= V by A5,TOPREAL9:13;
  end;
  assume
A6: for p st p in V holds ex r being Real st r > 0 & Ball(p,r) c= V;
  for x being Element of Euclid n st x in V1 holds
  ex r being Real st r > 0 & Ball(x,r) c= V1
  proof
    let x be Element of Euclid n;
    assume
A7: x in V1;
    reconsider p = x as Point of T by EUCLID:67;
    consider r be Real such that
A8: r > 0 & Ball(p,r) c= V by A6,A7;
    take r;
    thus r > 0 by A8;
    reconsider x1 = x as Point of Euclid n1;
    reconsider p1 = p as Point of TOP-REAL n1;
    thus Ball(x,r) c= V1 by A8,TOPREAL9:13;
  end;
  hence V in the topology of T by A1,A2,PCOMPS_1:def 4;
end;
