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
$$S=\bigcap _{m=1}^{\mathrm{\infty}}\bigcup _{n=1}^{\mathrm{\infty}}{A}_{m,n}\times {B}_{m,n}$$ 
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  20130322 18:46:30 
Last modified on  20130322 18:46:30 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  4 
Author  gel (22282) 
Entry type  Theorem 
Classification  msc 28A05 