Development (topology)

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 $$F_1, F_2, \ldots$$ of open coverings of $$X$$, such that for any closed subset $$C \subset X$$ 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 $$F_1, F_2,\ldots$$ such that $$F_{i+1}\subset F_i$$ 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.