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Hilbert cube; discrete-dense; disjoint; disconnected; covering; constructive; computation; digital interval; $T_4$-space
We effectively construct in the Hilbert cube $\Bbb H= [0,1]^\omega$ two sets $V, W \subset \Bbb H$ 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 $\Bbb H$. In fact, $V = \bigcup_{\Bbb N} V_i$, $W = \bigcup_{\Bbb N} W_i$, 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 $(0, 0, 0, \ldots) \in V_{ij}$, $(1, 1, 1, \ldots) \in W_{ij}$, (d) $V_i \cup W_i$, $i \in \Bbb N$ 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 $i$.
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