
theorem Th41:
  for V being RealUnitarySpace holds TopStruct (#the carrier of V,
    Family_open_set(V)#) is TopSpace
proof
  let V be RealUnitarySpace;
  set T = TopStruct (#the carrier of V,Family_open_set(V)#);
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 Th39;
  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 Th38,Th40;
  hence thesis by A1,PRE_TOPC:def 1;
end;
