# development

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 $C$ 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 $F_{1},F_{2},\ldots$ 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.

## References

• 1 Steen, Lynn Arthur and Seebach, J. Arthur, Counterexamples in Topology, Dover Books, 1995.
Title development Development 2013-03-22 14:49:49 2013-03-22 14:49:49 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 54D20 developable nested development Vickery’s theorem