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.
References
- 1 Steen, Lynn Arthur and Seebach, J. Arthur, Counterexamples in Topology, Dover Books, 1995.
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 |