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 [math]\displaystyle{ X }[/math] be a topological space. A development for [math]\displaystyle{ X }[/math] is a countable collection [math]\displaystyle{ F_1, F_2, \ldots }[/math] of open coverings of [math]\displaystyle{ X }[/math], such that for any closed subset [math]\displaystyle{ C \subset X }[/math] and any point [math]\displaystyle{ p }[/math] in the complement of [math]\displaystyle{ C }[/math], there exists a cover [math]\displaystyle{ F_j }[/math] such that no element of [math]\displaystyle{ F_j }[/math] which contains [math]\displaystyle{ p }[/math] intersects [math]\displaystyle{ C }[/math]. A space with a development is called developable.

A development [math]\displaystyle{ F_1, F_2,\ldots }[/math] such that [math]\displaystyle{ F_{i+1}\subset F_i }[/math] for all [math]\displaystyle{ i }[/math] is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If [math]\displaystyle{ F_{i+1} }[/math] is a refinement of [math]\displaystyle{ F_i }[/math], for all [math]\displaystyle{ i }[/math], 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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