development

Let $X$ be a topological space. A development for $X$ is a countable collection $F_1,F_2,\mathrm{\dots }$ of open coverings of $X$ such that for any closed subset $C$ of $X$ 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 $F_1,F_2,\mathrm{\dots }$ such that $F_i\subset F_{i+1}$ for all $i$ is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development.

Title	development
Canonical name	Development
Date of creation	2013-03-22 14:49:49
Last modified on	2013-03-22 14:49:49
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Definition
Classification	msc 54D20
Defines	developable
Defines	nested development
Defines	Vickery’s theorem