
theorem Th22:
  for T being non empty TopSpace, x being Element of Omega T holds
  Cl {x} = downarrow x
proof
  let T be non empty TopSpace, x be Element of Omega T;
A1: the TopStruct of T = the TopStruct of Omega T by Def2;
  hereby
    reconsider Z = {x} as Subset of T by A1;
    let a be object;
    assume
A2: a in Cl {x};
    then reconsider b = a as Element of Omega T;
    a in Cl Z by A1,A2,TOPS_3:80;
    then b <= x by Def2;
    hence a in downarrow x by WAYBEL_0:17;
  end;
  let a be object;
  assume
A3: a in downarrow x;
  then reconsider b = a as Element of Omega T;
  b <= x by A3,WAYBEL_0:17;
  then ex Z being Subset of T st Z = {x} & b in Cl Z by Def2;
  hence thesis by A1,TOPS_3:80;
end;
