reserve T for TopStruct;
reserve GX for TopSpace;
reserve T for TopStruct,
  x,y for Point of T;

theorem Th31:
  for T being TopSpace, X being set holds X is closed Subset of T
  iff X is closed Subset of the TopStruct of T
proof
  let T be TopSpace, X be set;
  [#]T \ X is open iff ([#]the TopStruct of T) \ X is open;
  hence thesis by Def3;
end;
