Development (topology) explained

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

• Book: Steen . Lynn Arthur . Lynn Arthur Steen . Seebach . J. Arthur Jr. . J. Arthur Seebach, Jr. . . 2nd . 1978 . . Berlin, New York . 3-540-90312-7 . 507446 . 0386.54001 .
• Vickery . C.W. . Axioms for Moore spaces and metric spaces . Bull. Amer. Math. Soc. . 46 . 1940 . 6 . 560–564 . 10.1090/S0002-9904-1940-07260-X . 0061.39807 . 66.0208.03 . free .