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

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

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