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Let $X$ be a topological space. A development for $X$ is a countable collection $F_1, F_2, \ldots$ of open coverings of $X$ such that for any closed subset $C$ of $X$ and any point $p$ in the complement of $C$ , there exists a cover $F_j$ such that no element of $F_j$ which contains $p$ intersects $C$ . A space with a development is called developable.
A development $F_1, F_2,\ldots$ such that $F_i\subset F_{i+1}$ for all $i$ is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development.
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- Steen, Lynn Arthur and Seebach, J. Arthur, Counterexamples in Topology, Dover Books, 1995.
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"development" is owned by mathcam.
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| Also defines: |
developable, nested development, Vickery's theorem |
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Cross-references: theorem, intersects, contains, element, cover, complement, point, closed subset, coverings, open, collection, countable, topological space
There are 56 references to this entry.
This is version 3 of development, born on 2004-11-18, modified 2004-11-19.
Object id is 6495, canonical name is Development.
Accessed 6607 times total.
Classification:
| AMS MSC: | 54D20 (General topology :: Fairly general properties :: Noncompact covering properties ) |
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Pending Errata and Addenda
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