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 507446. 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.
• This article incorporates material from Development on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.