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### Filling boxes densely and disjointly

Commentationes Mathematicae Universitatis Carolinae

We effectively construct in the Hilbert cube $ℍ={\left[0,1\right]}^{\omega }$ two sets $V,W\subset ℍ$ with the following properties: (a) $V\cap W=\varnothing$, (b) $V\cup W$ is discrete-dense, i.e. dense in ${{\left[0,1\right]}_{D}}^{\omega }$, where ${\left[0,1\right]}_{D}$ denotes the unit interval equipped with the discrete topology, (c) $V$, $W$ are open in $ℍ$. In fact, $V={\bigcup }_{ℕ}{V}_{i}$, $W={\bigcup }_{ℕ}{W}_{i}$, 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 $\left(0,0,0,...\right)\in {V}_{ij}$, $\left(1,1,1,...\right)\in {W}_{ij}$, (d) ${V}_{i}\cup {W}_{i}$, $i\in ℕ$ is point symmetric about $\left(1/2,1/2,1/2,...\right)$. Instead of $\left[0,1\right]$ 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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