A paving on a set is any collection of subsets of , and is said to be a paved space. Given any two paved spaces and , the product paving is defined as
A paved space is said to be compact if every subcollection of satisfying the finite intersection property has nonempty intersection. Equivalently, if any has empty intersection then there is a finite with empty intersection. Then, is said to be a compact paving, and is compactly paved by . An example of compact pavings is given by the collection of all compact subsets (http://planetmath.org/Compact) of a Hausdorff topological space.
Similarly, denotes the countable intersections of elements of ,
These operations can be combined in any order so that, for example, is the collection of countable intersections of countable unions of elements of .
Note: In the definition of a paved space, some authors additionally require a paving to contain the empty set.
- 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.
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|Date of creation||2013-03-22 18:44:50|
|Last modified on||2013-03-22 18:44:50|
|Last modified by||gel (22282)|
|Defines||compactly paved by|