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 , … of open coverings of X , such that for any closed subset C ⊂ X and any point p in the complement of C , there exists a cover F j such that no element of F j which contains p intersects C . A space with a development is called developable.
A development F 1 , F 2 , … such that F i + 1 ⊂ 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 F i + 1 is a refinement of F i , 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.
