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

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

• 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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