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 <math>X</math> be a topological space. A development for <math>X</math> is a countable collection <math>F_1, F_2, \ldots</math> of open coverings of <math>X</math>, such that for any closed subset <math>C \subset X</math> and any point <math>p</math> in the complement of <math>C</math>, there exists a cover <math>F_j</math> such that no element of <math>F_j</math> which contains <math>p</math> intersects <math>C</math>. A space with a development is called developable.

A development <math>F_1, F_2,\ldots</math> such that <math>F_{i+1}\subset F_i</math> for all <math>i</math> is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If <math>F_{i+1}</math> is a refinement of <math>F_i</math>, for all <math>i</math>, 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.