# Filling boxes densely and disjointly

• Volume: 44, Issue: 1, page 187-196
• ISSN: 0010-2628

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## Abstract

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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 $i$.

## How to cite

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Schröder, J.. "Filling boxes densely and disjointly." Commentationes Mathematicae Universitatis Carolinae 44.1 (2003): 187-196. <http://eudml.org/doc/249177>.

@article{Schröder2003,
abstract = {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 $i$.},
author = {Schröder, J.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Hilbert cube; discrete-dense; disjoint; disconnected; covering; constructive; computation; digital interval; $T_4$-space; Hilbert cube; discrete-dense; disjoint; disconnected; covering; constructive; computation; digital interval; -space},
language = {eng},
number = {1},
pages = {187-196},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Filling boxes densely and disjointly},
url = {http://eudml.org/doc/249177},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Schröder, J.
TI - Filling boxes densely and disjointly
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 1
SP - 187
EP - 196
AB - 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 $i$.
LA - eng
KW - Hilbert cube; discrete-dense; disjoint; disconnected; covering; constructive; computation; digital interval; $T_4$-space; Hilbert cube; discrete-dense; disjoint; disconnected; covering; constructive; computation; digital interval; -space
UR - http://eudml.org/doc/249177
ER -

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