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

A development F 1 , F 2 , … {\displaystyle F_{1},F_{2},\ldots } such that F i + 1 ⊂ F i {\displaystyle F_{i+1}\subset F_{i}} for all i {\displaystyle 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 {\displaystyle F_{i+1}} is a refinement of F i {\displaystyle F_{i}}, for all i {\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.