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 and Seebach, J. Arthur, Counterexamples in Topology, Dover Books, 1995.
• Vickery, C.W. Axioms for Moore spaces and metric spaces. Bull. Amer. Math. Soc., 46 (1940), 560-564.
• This article incorporates material from Development on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.