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| In the [[mathematical]] field of [[topology]], a '''development''' is a [[countable]] collection of [[open cover]]s of a [[topological space]] that satisfies certain [[separation axiom]]s.
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| 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 (set theory)|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'''.
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| 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 (topology)|refinement]] of <math>F_i</math>, for all <math>i</math>, then the development is called a '''refined development'''.
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| Vickery's theorem implies that a topological space is a [[Moore space (topology)|Moore space]] if and only if it is [[regular space|regular]] and developable.
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| ==References==
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| * Steen, Lynn Arthur and Seebach, J. Arthur, ''Counterexamples in Topology'', Dover Books, 1995.
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| * Vickery, C.W. ''Axioms for Moore spaces and metric spaces''. Bull. Amer. Math. Soc., 46 (1940), 560-564.
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| * {{PlanetMath attribution|id=6495|title=Development}}
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| [[Category:General topology]]
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Latest revision as of 02:40, 8 December 2014
Hello, my title is Andrew and my wife doesn't like it at all. She functions as a journey agent but soon she'll be on her personal. Alaska is exactly where he's usually been residing. To climb is something I truly appreciate doing.
Have a look at my site: best psychics (http://Bigpolis.com)