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

1.
$A$ is $\mathcal{F}$analytic.

2.
There is a closed subset $S$ of $\mathcal{N}$ and $\theta :{\mathbb{N}}^{2}\to \mathcal{F}$ such that
$$A=\bigcup _{s\in S}\bigcap _{n=1}^{\mathrm{\infty}}\theta (n,{s}_{n}).$$ 
3.
There is a closed subset $S$ of $\mathcal{N}$ and $\theta :\mathbb{N}\to \mathcal{F}$ such that
$$A=\bigcup _{s\in S}\bigcap _{n=1}^{\mathrm{\infty}}\theta \left({s}_{n}\right).$$ 
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 $\mathrm{(}X\mathrm{,}\mathrm{F}\mathrm{)}$ 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^{} $f:Z\to X$ for some Polish space $Z$.

5.
$A$ is the image of a continuous function $f:\mathcal{N}\to X$.

6.
$A$ is the image of a Borel measurable function $f:Y\to X$.
Title  equivalent definitions of analytic sets 

Canonical name  EquivalentDefinitionsOfAnalyticSets 
Date of creation  20130322 18:48:28 
Last modified on  20130322 18:48:28 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  6 
Author  gel (22282) 
Entry type  Theorem 
Classification  msc 28A05 