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 $F\_1,\; F\_2,\; \backslash ldots$ of open coverings of $X$, such that for any closed subset $C\; \backslash subset\; X$ 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 $F\_1,\; F\_2,\backslash ldots$ such that $F\_\{i+1\}\backslash subset\; F\_i$ 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.