reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;
reserve T for TopSpace;
reserve A for Subset of T;
reserve T for non empty TopSpace;
reserve x for Point of T;
reserve Z,X,V,W,Y,Q for Subset of T;
reserve FX for Subset-Family of T;
reserve a for set;
reserve x,y for Point of T;
reserve A,B for Subset of T;
reserve FX,GX for Subset-Family of T;
reserve x,y,z for Element of PM;
reserve V,W,Y for Subset of PM;

theorem Th32:
  for A being Subset-Family of PM st A c= Family_open_set(PM)
  holds union A in Family_open_set(PM)
proof
  let A be Subset-Family of PM;
  assume
A1: A c= Family_open_set(PM);
  for x st x in union A ex r st r>0 & Ball(x,r) c= union A
  proof
    let x;
    assume x in union A;
    then consider W being set such that
A2: x in W and
A3: W in A by TARSKI:def 4;
    reconsider W as Subset of PM by A3;
    consider r such that
A4: r>0 and
A5: Ball(x,r) c= W by A1,A2,A3,Def4;
    take r;
    thus r > 0 by A4;
    W c= union A by A3,ZFMISC_1:74;
    hence thesis by A5;
  end;
  hence thesis by Def4;
end;
