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. 1.

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

2. 2.

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

 $A=\bigcup_{s\in S}\bigcap_{n=1}^{\infty}\theta\left(n,s_{n}\right).$
3. 3.

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. 4.

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

5. 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. 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. 1.

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

2. 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

Title equivalent definitions of analytic sets EquivalentDefinitionsOfAnalyticSets 2013-03-22 18:48:28 2013-03-22 18:48:28 gel (22282) gel (22282) 6 gel (22282) Theorem msc 28A05