Analytic set

Remarks

Every closed subset $F$ of $X$ is analytic: if $F$ is nonempty, it is a Polish space, and every nonempty Polish space is a continuous image of $\mathcal{N}$. More generally every Borel subset of $X$ is analytic. Non-example by counting: in $X = \R$ there are only $2^{\aleph_0}$ analytic sets. Indeed, each nonempty one is the image of a continuous map on $\mathcal{N}$, and such a map is determined by its values on a countable dense set. Since $\R$ has $2^{2^{\aleph_0}}$ subsets, some subset of $\R$ is not analytic. The following are equivalent forms of the definition. $A$ is the projection to $X$ of a closed subset of $X \times \mathcal{N}$. $A$ is the projection to $X$ of a Borel subset of $X \times Y$ for some Polish space $Y$. $A$ is empty or a continuous image of a Polish space. Complements of analytic sets are called coanalytic, and they form the class $\Pi^1_1$.