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 ${\displaystyle X}$ be a topological space. A development for ${\displaystyle X}$ is a countable collection ${\displaystyle F_{1},F_{2},\ldots }$ of open coverings of ${\displaystyle X}$, such that for any closed subset ${\displaystyle C\subset X}$ and any point ${\displaystyle p}$ in the complement of ${\displaystyle C}$, there exists a cover ${\displaystyle F_{j}}$ such that no element of ${\displaystyle F_{j}}$ which contains ${\displaystyle p}$ intersects ${\displaystyle C}$. A space with a development is called developable.

A development ${\displaystyle F_{1},F_{2},\ldots }$ such that ${\displaystyle F_{i+1}\subset F_{i}}$ for all ${\displaystyle i}$ is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If ${\displaystyle F_{i+1}}$ is a refinement of ${\displaystyle F_{i}}$, for all ${\displaystyle i}$, 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.