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 Th30:
  the carrier of PM in Family_open_set(PM)
proof
  the carrier of PM c= the carrier of PM & for y st y in the carrier of PM
  holds ex p st p>0 & Ball(y,p) c= the carrier of PM by Th26;
  hence thesis by Def4;
end;
