Development (topology)

In the mathematical field of topology, a development is a countable collection of open covers of a topological space that satisfies certain separation axioms.

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 $C\subset X$ 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 $F_{i+1}\subset F_{i}$ for all $i$ is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If $F_{{i+1}}$ is a refinement of $F_{i}$ , for all $i$ , then the development is called a refined development.

Vickery's theorem implies that a topological space is a Moore space if and only if it is regular and developable.

References

Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 3-540-90312-7. MR 0507446. Zbl 0386.54001.
Vickery, C.W. (1940). "Axioms for Moore spaces and metric spaces". Bull. Amer. Math. Soc. 46: 560–564. doi:10.1090/S0002-9904-1940-07260-X. JFM 66.0208.03. Zbl 0061.39807.