analytic sets define a closure operator

For a paving $\mathcal{F}$ on a set $X$ , we denote the collection of all $\mathcal{F}$ -analytic sets (http://planetmath.org/AnalyticSet2) by $a(\mathcal{F})$ . Then, $\mathcal{F}\mapsto a(\mathcal{F})$ is a closure operator on the subsets of $X$ . That is,

1.

$\mathcal{F}\subseteq a(\mathcal{F})$ .
2.

If $\mathcal{F}\subseteq\mathcal{G}$ then $a(\mathcal{F})\subseteq a(\mathcal{G})$ .
3.

$a(a(\mathcal{F}))=a(\mathcal{F})$ .

For example, if $\mathcal{G}$ is a collection of $\mathcal{F}$ -analytic sets then $\mathcal{G}\subseteq a(\mathcal{F})$ gives $a(\mathcal{G})\subseteq a(a(\mathcal{F}))=a(\mathcal{F})$ and so all $\mathcal{G}$ -analytic sets are also $\mathcal{F}$ -analytic. In particular, for a metric space, the analytic sets are the same regardless of whether they are defined with respect to the collection of open, closed or Borel sets.

Properties 1 and 2 follow directly from the definition of analytic sets. We just need to prove 3. So, for any $A\in a(a(\mathcal{F}))$ we show that $A\in a(\mathcal{F})$ . First, there is a compact paved space (http://planetmath.org/PavedSpace) $(K,\mathcal{K})$ and $S\in\left(a(\mathcal{F})\times\mathcal{K}\right)_{\sigma\delta}$ such that $A$ is equal to the projection $\pi_{X}(S)$ . Write

S=\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty}A_{m,n}\times B_{m,n}

for $A_{m,n}\in a(\mathcal{F})$ and $B_{m,n}\in\mathcal{K}$ . It is clear that $A_{m,n}\times B_{m,n}$ is $\mathcal{F}\times\mathcal{K}$ -analytic and, as countable unions and intersections of analytic sets are analytic, $S$ is also $\mathcal{F}\times\mathcal{K}$ -analytic. Finally, since projections of analytic sets are analytic, $A=\pi_{X}(S)$ must be $\mathcal{F}$ -analytic as required.

Title	analytic sets define a closure operator
Canonical name	AnalyticSetsDefineAClosureOperator
Date of creation	2013-03-22 18:46:30
Last modified on	2013-03-22 18:46:30
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	4
Author	gel (22282)
Entry type	Theorem
Classification	msc 28A05