development
Development
2004-11-18 21:41:06
2004-11-19 11:55:42
Definition
developable
nested development
Vickery's theorem
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Let $X$ be a topological space. A \emph{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 \emph{developable}.
A development $F_1, F_2,\ldots$ such that $F_i\subset F_{i+1}$ for all $i$ is called a \emph{nested development}. A theorem from Vickery states that every developable space in fact has a nested development.
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\bibitem{a}
Steen, Lynn Arthur and Seebach, J. Arthur, \emph{Counterexamples in Topology}, Dover Books, 1995.
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