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 F1,F2,… of open coverings of X, such that for any closed subset C⊂X and any point p in the complement of C, there exists a cover Fj such that no element of Fj which contains p intersects C. A space with a development is called developable.

A development F1,F2,… such that Fi+1⊂Fi for all i is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If Fi+1 is a refinement of Fi, 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.
* Vickery, C.W. (1940). "Axioms for Moore spaces and metric spaces". Bull. Amer. Math. Soc. 46 (6): 560–564. doi:10.1090/S0002-9904-1940-07260-X.

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