# paved space

A paving on a set $X$ is any collection $\mathcal{A}$ of subsets of $X$, and $(X,\mathcal{A})$ is said to be a paved space. Given any two paved spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$, the product paving $\mathcal{A}\times\mathcal{B}$ is defined as

 $\mathcal{A}\times\mathcal{B}=\left\{A\times B\colon A\in\mathcal{A},B\in% \mathcal{B}\right\}.$

A paved space $(K,\mathcal{K})$ is said to be compact if every subcollection of $\mathcal{K}$ satisfying the finite intersection property has nonempty intersection. Equivalently, if any $\mathcal{K}^{\prime}\subseteq\mathcal{K}$ has empty intersection then there is a finite $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K}^{\prime}$ with empty intersection. Then, $\mathcal{K}$ is said to be a compact paving, and $K$ is compactly paved by $\mathcal{K}$. An example of compact pavings is given by the collection of all compact subsets (http://planetmath.org/Compact) of a Hausdorff topological space.

For any paving $\mathcal{A}$, the notation $\mathcal{A}_{\sigma}$ is often used to denote countable unions of elements of $\mathcal{A}$,

 $\mathcal{A}_{\sigma}\equiv\left\{\bigcup_{n=1}^{\infty}A_{n}\colon A_{n}\in% \mathcal{A}\text{ for all }n\in\mathbb{N}\right\}.$

Similarly, $\mathcal{A}_{\delta}$ denotes the countable intersections of elements of $\mathcal{A}$,

 $\mathcal{A}_{\delta}\equiv\left\{\bigcap_{n=1}^{\infty}A_{n}\colon A_{n}\in% \mathcal{A}\text{ for all }n\in\mathbb{N}\right\}.$

These operations can be combined in any order so that, for example, $\mathcal{A}_{\sigma\delta}=(\mathcal{A}_{\sigma})_{\delta}$ is the collection of countable intersections of countable unions of elements of $\mathcal{A}$.

Note: In the definition of a paved space, some authors additionally require a paving $\mathcal{K}$ to contain the empty set.

## References

• 1 K. Bichteler, Stochastic integration with jumps. Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, 2002.
• 2 Claude Dellacherie, Paul-André Meyer, Probabilities and potential. North-Holland Mathematics Studies, 29. North-Holland Publishing Co., 1978.
• 3 Sheng-we He, Jia-gang Wang, Jia-an Yan, Semimartingale theory and stochastic calculus. Kexue Chubanshe (Science Press), CRC Press, 1992.
• 4 M.M. Rao, Measure theory and integration. Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 265. Marcel Dekker Inc., 2004.
 Title paved space Canonical name PavedSpace Date of creation 2013-03-22 18:44:50 Last modified on 2013-03-22 18:44:50 Owner gel (22282) Last modified by gel (22282) Numerical id 6 Author gel (22282) Entry type Definition Classification msc 28A05 Synonym paving Synonym paved set Related topic F_sigmaSet Related topic G_deltaSet Related topic AnalyticSet2 Defines paving Defines compact paving Defines compactly paved by Defines product paving