paved space

A paving on a set X is any collectionMathworldPlanetmath π’œ of subsets of X, and (X,π’œ) is said to be a paved space. Given any two paved spaces (X,π’œ) and (Y,ℬ), the product paving π’œΓ—β„¬ is defined as


A paved space (K,𝒦) is said to be compactPlanetmathPlanetmath if every subcollection of 𝒦 satisfying the finite intersection property has nonempty intersectionDlmfMathworldPlanetmath. Equivalently, if any π’¦β€²βŠ†π’¦ has empty intersection then there is a finite π’¦β€²β€²βŠ†π’¦β€² with empty intersection. Then, 𝒦 is said to be a compact paving, and K is compactly paved by 𝒦. An example of compact pavings is given by the collection of all compact subsets ( of a Hausdorff topological space.

For any paving π’œ, the notation π’œΟƒ is often used to denote countableMathworldPlanetmath unions of elements of π’œ,

π’œΟƒβ‰‘{⋃n=1∞An:Anβˆˆπ’œβ’Β for all ⁒nβˆˆβ„•}.

Similarly, π’œΞ΄ denotes the countable intersections of elements of π’œ,

π’œΞ΄β‰‘{β‹‚n=1∞An:Anβˆˆπ’œβ’Β for all ⁒nβˆˆβ„•}.

These operationsMathworldPlanetmath can be combined in any order so that, for example, π’œΟƒβ’Ξ΄=(π’œΟƒ)Ξ΄ is the collection of countable intersections of countable unions of elements of π’œ.

Note: In the definition of a paved space, some authors additionally require a paving 𝒦 to contain the empty setMathworldPlanetmath.


  • 1 K. Bichteler, Stochastic integration with jumps. Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, 2002.
  • 2 Claude Dellacherie, Paul-AndrΓ© 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