equivalent definitions of analytic sets

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

In conditions 2 and 3 of the following theorem, Baire space $𝒩={\mathbb{N}}^{\mathbb{N}}$ 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 $Y=ℝ$ 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 $ℱ$-analytic.

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

   $$A=\bigcup _{s\in S}\bigcap _{n=1}^{\mathrm{\infty }}\theta (n,s_n).$$

3. There is a closed subset $S$ of $𝒩$ and $\theta :\mathbb{N}\to ℱ$ 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 $ℱ$.

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

6. $A$ is the projection of a set in ${(ℱ\times 𝒦)}_{\sigma \delta }$ onto $X$, where $𝒦$ 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 $ℬ$. 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 ${ℝ}^2$.

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 $ℱ$-analytic.

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

We finally state some equivalent definitions of analytic subsets of a Polish space. Again, $𝒩$ 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 $ℱ$-analytic (http://planetmath.org/AnalyticSet2).

2. $A$ is the projection of a closed subset of $X\times 𝒩$ 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 $f:𝒩\to X$.

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