J. Schr\"oder Filling boxes densely and disjointly

Comment.Math.Univ.Carolinae 44,1 (2003) 187-196.

Abstract:We effectively construct in the Hilbert cube $\H = [0,1]^\omega$ two sets $V, W \subset \H$ with the following properties: \roster \item "(a)" $V \cap W = \emptyset$, \item "(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, \item "(c)" $V$,$W$ are open in $\H$. In fact, $V = \bigcup _{\relax$\Bbb N$} V_i$, $W = \bigcup _{\relax$\Bbb 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 \newline $(0, 0, 0, \ldots ) \in V_{ij}$, $(1, 1, 1, \ldots ) \in W_{ij}$, \item "(d)" $V_i \cup W_i$, $i \in \relax$\Bbb N is point symmetric about $(1/2, 1/2, 1/2, \ldots )$. \endroster Instead of $[0,1]$ we could have taken any $T_4$-space or a digital interval, where the resolution (number of points) increases with $i$.

Keywords: Hilbert cube, discrete-dense, disjoint, disconnected, covering, constructive, computation, digital interval, $T_4$-space
AMS Subject Classification: Primary 54-04; Secondary 05-04, 54B10