equivalent definitions of analytic sets

For a paved space $(X,\mathcal{F})$ the $\mathcal{F}$ -analytic (http://planetmath.org/AnalyticSet2) sets can be defined as the projections (http://planetmath.org/GeneralizedCartesianProduct) of sets in $(\mathcal{F}\times\mathcal{K})_{\sigma\delta}$ onto $X$ , for compact paved spaces $(K,\mathcal{K})$ . There are, however, many other equivalent definitions, some of which we list here.

In conditions 2 and 3 of the following theorem, Baire space $\mathcal{N}=\mathbb{N}^{\mathbb{N}}$ is the collection of sequences of natural numbers together with the product topology. In conditions 5 and 6, $Y$ can be any uncountable Polish space. For example, we may take $Y=\mathbb{R}$ with the standard topology.

Theorem.

Let $(X,\mathcal{F})$ be a paved space such that $\mathcal{F}$ contains the empty set, and $A$ be a subset of $X$ . The following are equivalent.

1.

$A$ is $\mathcal{F}$ -analytic.

There is a closed subset $S$ of $\mathcal{N}$ and $\theta\colon\mathbb{N}^{2}\to\mathcal{F}$ such that

$A=\bigcup_{s\in S}\bigcap_{n=1}^{\infty}\theta\left(n,s_{n}\right).$

There is a closed subset $S$ of $\mathcal{N}$ and $\theta\colon\mathbb{N}\to\mathcal{F}$ such that

$A=\bigcup_{s\in S}\bigcap_{n=1}^{\infty}\theta\left(s_{n}\right).$

4.

$A$ is the result of a Souslin scheme on $\mathcal{F}$ .
5.

$A$ is the projection of a set in $(\mathcal{F}\times\mathcal{G})_{\sigma\delta}$ onto $X$ , where $\mathcal{G}$ is the collection of closed subsets of $Y$ .
6.

$A$ is the projection of a set in $(\mathcal{F}\times\mathcal{K})_{\sigma\delta}$ onto $X$ , where $\mathcal{K}$ is the collection of compact subsets of $Y$ .

For subsets of a measurable space, the following result gives a simple condition to be analytic. Again, the space $Y$ can be any uncountable Polish space, and its Borel $\sigma$ -algebra is denoted by $\mathcal{B}$ . In particular, this result shows that a subset of the real numbers is analytic if and only if it is the projection of a Borel set from $\mathbb{R}^{2}$ .

Theorem.

Let $(X,\mathcal{F})$ be a measurable space. For a subset $A$ of $X$ the following are equivalent.

1.

$A$ is $\mathcal{F}$ -analytic.
2.

$A$ is the projection of an $\mathcal{F}\otimes\mathcal{B}$ -measurable subset of $X\times Y$ onto $X$ .

We finally state some equivalent definitions of analytic subsets of a Polish space. Again, $\mathcal{N}$ denotes Baire space and $Y$ is any uncountable Polish space.

Theorem.

For a nonempty subset $A$ of a Polish space $X$ the following are equivalent.

1.

$A$ is $\mathcal{F}$ -analytic (http://planetmath.org/AnalyticSet2).
2.

$A$ is the projection of a closed subset of $X\times\mathcal{N}$ onto $X$ .
3.

$A$ is the projection of a Borel subset of $X\times Y$ onto $X$ .
4.

$A$ is the image (http://planetmath.org/DirectImage) of a continuous function $f\colon Z\to X$ for some Polish space $Z$ .
5.

$A$ is the image of a continuous function $f\colon\mathcal{N}\to X$ .
6.

$A$ is the image of a Borel measurable function $f\colon Y\to X$ .

Title	equivalent definitions of analytic sets
Canonical name	EquivalentDefinitionsOfAnalyticSets
Date of creation	2013-03-22 18:48:28
Last modified on	2013-03-22 18:48:28
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	6
Author	gel (22282)
Entry type	Theorem
Classification	msc 28A05