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 Th33:
  TopStruct (#the carrier of PM,Family_open_set(PM)#) is TopSpace
proof
  set T = TopStruct (#the carrier of PM,Family_open_set(PM)#);
A1: for p,q being Subset of T st p in the topology of T & q in the topology
  of T holds p /\ q in the topology of T by Th31;
  the carrier of T in the topology of T & for a being Subset-Family of T
  st a c= the topology of T holds union a in the topology of T by Th30,Th32;
  hence thesis by A1,PRE_TOPC:def 1;
end;
