Development (topology)

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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 Fj such that no element of Fj 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 Fi + 1 is a refinement of Fi, 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.

[edit] 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 GFDL.