Filling boxes densely and disjointly

We effectively construct in the Hilbert cube H = [0, 1]ω two sets V, W ⊂ H with the following properties: (a) V ∩ W = ∅, (b) V ∪ W is discrete-dense, i.e. dense in [0, 1]D ω , where [0, 1]D denotes the unit interval equipped with the discrete topology, (c) V , W are open in H. In fact, V = S N Vi, W = S N Wi, where Vi = S2i−1−1 0 Vij , Wi = S2i−1−1 0 Wij . Vij , Wij are basic open sets and (0, 0, 0, . . . ) ∈ Vij , (1, 1, 1, . . . ) ∈ Wij , (d) Vi ∪ Wi, i ∈ N is point symmetric about (1/2, 1/2, 1/2, . . . ). Instead of [0, 1] we could have taken any T4-space or a digital interval, where the resolution (number of points) increases with i.