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 be a topological space. A development for is a countable collection of open coverings of , such that for any closed subset and any point in the complement of , there exists a cover such that no element of which contains intersects . A space with a development is called developable.

A development such that for all is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If is a refinement of , for all , 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.