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 Th24:
  for O being open non empty Subset of TOP-REAL n holds
  ex Y being Subset of dense_countable_OpenHypercubes(n) st Y is non empty &
    O = union Y &
  ex g being Function of NAT,Y st
  for x being object holds x in O iff ex y being object st y in NAT & x in g.y
  proof
    let O be open non empty Subset of TOP-REAL n;
    consider Y be Subset of dense_countable_OpenHypercubes(n) such that
    Y is countable and
A1: O = union Y by Th23;
    take Y;
    thus Y is non empty by A1,ZFMISC_1:2;
    thus O = union Y by A1;
    consider g be Function of omega,Y such that
A2: rng g = Y by A1,ZFMISC_1:2,CARD_3:96;
    take g;
A3: dom g = omega by A1,ZFMISC_1:2,FUNCT_2:def 1;
    O = Union g by A1,A2;
    hence thesis by A3,CARD_5:2;
  end;
