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 be a topological space. A development for is a countable collection of open coverings of , such that for any closed subset and any point in the complement of , there exists a cover such that no element of which contains intersects . A space with a development is called developable.
A development such that for all is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If is a refinement of , for all , 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.
- This article incorporates material from Development on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.