reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;
reserve Un for FamilySequence of T,
  r,r1,r2 for Real,
  n for Element of NAT;

theorem Th14:
  for r being Real, PM be non empty MetrSpace for x being
  Element of PM holds [#]PM\cl_Ball(x,r) in Family_open_set(PM)
proof
  let r be Real;
  reconsider r9=r as Real;
  let PM be non empty MetrSpace;
  let x be Element of PM;
  now
    let y be Element of PM;
    set r1=dist(x,y)-r9;
A1: Ball(y,r1) c= [#]PM\cl_Ball(x,r)
    proof
      assume not Ball(y,r1) c= [#]PM\cl_Ball(x,r);
      then consider z being object such that
A2:   z in Ball(y,r1) and
A3:   not z in [#]PM\cl_Ball(x,r);
      reconsider z as Element of PM by A2;
      not (z in [#]PM & not z in cl_Ball(x,r)) by A3,XBOOLE_0:def 5;
      then
A4:   dist(x,z)<=r9 by METRIC_1:12;
      dist(y,z)<r1 by A2,METRIC_1:11;
      then dist(y,z) + dist(x,z)< r1 + r9 by A4,XREAL_1:8;
      then dist(x,z)+dist(z,y)< r1 + r9;
      hence thesis by METRIC_1:4;
    end;
    assume y in [#]PM\cl_Ball(x,r);
    then not y in cl_Ball(x,r) by XBOOLE_0:def 5;
    then r9+0<r9+r1 by METRIC_1:12;
    then r9-r9<r1-0 by XREAL_1:21;
    hence ex r2 st r2>0 & Ball(y,r2) c= [#]PM\cl_Ball(x,r) by A1;
  end;
  hence thesis by PCOMPS_1:def 4;
end;
