Article
| Title: | Filling boxes densely and disjointly (English) |
| Author: | Schröder, J. |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 44 |
| Issue: | 1 |
| Year: | 2003 |
| Pages: | 187-196 |
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| Category: | math |
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| Summary: | 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 $i$. (English) |
| Keyword: | Hilbert cube |
| Keyword: | discrete-dense |
| Keyword: | disjoint |
| Keyword: | disconnected |
| Keyword: | covering |
| Keyword: | constructive |
| Keyword: | computation |
| Keyword: | digital interval |
| Keyword: | $T_4$-space |
| MSC: | 05-04 |
| MSC: | 54-04 |
| MSC: | 54B10 |
| idZBL: | Zbl 1099.54011 |
| idMR: | MR2045855 |
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| Date available: | 2009-01-08T19:28:34Z |
| Last updated: | 2012-04-30 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/119377 |
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| Reference: | [Sch98] Schröder J.: On sub-, pseudo- and quasimaximal spaces.Comment. Math. Univ. Carolinae 39.1 (1998), 198-206. MR 1623022 |
| Reference: | [Wat90] Watson St.: Powers of the Sierpinski space.Topology Appl. 35 (1990), 299-302. Zbl 0698.54013, MR 1058809 |
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| Files | Size | Format | View |
|---|---|---|---|
| CommentatMathUnivCarolRetro_44-2003-1_15.pdf | 406.1Kb | application/pdf |
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