reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;

theorem Th19:
  union OpenHypercubesRAT(n) is open Subset-Family of TOP-REAL n
  proof
    reconsider UO = union OpenHypercubesRAT(n) as Subset-Family of TOP-REAL n
      by Th18;
    UO is open
    proof
      let P be Subset of TOP-REAL n;
      assume P in UO;
      then consider X be set such that
A1:   P in X and
A2:   X in OpenHypercubesRAT(n) by TARSKI:def 4;
      consider q0 be Point of Euclid n such that
A3:   X = OpenHypercubes(q0) and
      q0 in RAT n by A2;
A4:   OpenHypercubes(q0) is open;
      the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
      then reconsider P0=P as Subset of TopSpaceMetr Euclid n;
      P0 is open by A4,A3,A1;
      hence thesis by Th10;
    end;
    hence thesis;
  end;
