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 Th23:
  for O being open Subset of TOP-REAL n holds
  ex Y being Subset of dense_countable_OpenHypercubes(n) st Y is countable &
    O = union Y
  proof
    let O be open Subset of TOP-REAL n;
    the topology of TOP-REAL n c= UniCl dense_countable_OpenHypercubes(n)
      by Th21,CANTOR_1:def 2;
    then O in UniCl dense_countable_OpenHypercubes(n) by PRE_TOPC:def 2;
    then consider Y be Subset-Family of TOP-REAL n such that
A1: Y c= dense_countable_OpenHypercubes(n) and
A2: O = union Y by CANTOR_1:def 1;
    thus thesis by A1,A2;
  end;
