Filling boxes densely and disjointly. (English) Zbl 1099.54011

Schröder, J.

Commentat. Math. Univ. Carol. 44, No. 1, 187-196 (2003).

Summary: We effectively construct in the Hilbert cube \(\mathbb H = [0,1]^\omega \) two sets \(V, W \subset \mathbb H \) with the following properties:
(a) \(V \cap W = \emptyset \),
(b) \(V \cup W\) is discrete-dense, i.e. dense in \({[0,1]_D}^\omega \), where \([0,1]_D\) denotes the unit interval equipped with the discrete topology,
(c) \(V\), \(W\) are open in \(\mathbb H\). In fact, \(V = \bigcup _{\mathbb N}V_i\), \(W = \bigcup _{\mathbb N}W_i\), where \(V_i =\bigcup _0^{2^{i-1}-1}V_{ij}\), \(W_i = \bigcup _0^{2^{i-1}-1}W_{ij}\). \(V_{ij}\), \(W_{ij}\) are basic open sets and \((0, 0, 0, \ldots ) \in V_{ij}\), \((1, 1, 1, \ldots ) \in W_{ij}\),
(d) \(V_i \cup W_i\), \(i \in \mathbb N\) is point symmetric about \((1/2, 1/2, 1/2, \ldots )\).
Instead of \([0,1]\) we could have taken any \(T_4\)-space or a digital interval, where the resolution (number of points) increases with \(i\).

MSC:

54B10	Product spaces in general topology
54-04	Software, source code, etc. for problems pertaining to general topology
05-04	Software, source code, etc. for problems pertaining to combinatorics

Keywords:

Hilbert cube; discrete-dense; disjoint; disconnected; covering; constructive; computation; digital interval; \(T_4\)-space