paved space

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

$$𝒜\times ℬ=\{A\times B:A\in 𝒜,B\in ℬ\}.$$

A paved space $(K,𝒦)$ is said to be compact if every subcollection of $𝒦$ satisfying the finite intersection property has nonempty intersection. Equivalently, if any ${𝒦}^{\prime }\subseteq 𝒦$ has empty intersection then there is a finite ${𝒦}^{\prime \prime }\subseteq {𝒦}^{\prime }$ 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 (http://planetmath.org/Compact) of a Hausdorff topological space.

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

$${𝒜}_{\sigma }\equiv \{\bigcup _{n=1}^{\mathrm{\infty }}{A}_n:A_n\in 𝒜\text{ for all }n\in \mathbb{N}\}.$$

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

$${𝒜}_{\delta }\equiv \{\bigcap _{n=1}^{\mathrm{\infty }}{A}_n:A_n\in 𝒜\text{ for all }n\in \mathbb{N}\}.$$

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

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