
theorem Th23:
  for X being TopSpace, A being open Subset of Omega X holds A is upper
proof
  let X be TopSpace, A be open Subset of Omega X;
  let x, y be Element of Omega X such that
A1: x in A;
  assume x <= y;
  then
A2: ex Z being Subset of X st Z = {y} & x in Cl Z by Def2;
  then reconsider X as non empty TopSpace;
  reconsider y as Element of Omega X;
  the TopStruct of X = the TopStruct of Omega X by Def2;
  then x in Cl {y} by A2,TOPS_3:80;
  then A meets {y} by A1,PRE_TOPC:def 7;
  hence thesis by ZFMISC_1:50;
end;
