## Filling boxes densely and disjointly.(English)Zbl 1099.54011

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
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