Analytic and coanalytic sets by closed projection

Let $X$ be a Polish space (Polish spaces are separable completely metrizable spaces), and let $\mathcal{N}$ denote Baire sequence space ${\mathbb{N}}^{\mathbb{N}}$ and its cylinder topology. Use the binary product topology (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space) on $X\times \mathcal{N}$.

A set $A\subseteq X$ is analytic if there is a closed $F\subseteq X\times \mathcal{N}$ such that

$A=\{x\in X:(\mathrm{\exists }y\in \mathcal{N})(x,y)\in F\}\mathrm{.}$

A set $B\subseteq X$ is coanalytic if $X\setminus B$ is analytic. Complements are always relative to this specified $X$. The empty closed witness makes $\mathrm{\varnothing }$ analytic. The witness $X\times \mathcal{N}$ makes $X$ analytic: the constant-zero sequence witnesses the projection at each $x$. Hence both $\mathrm{\varnothing }$ and $X$ are also coanalytic, including when $X=\mathrm{\varnothing }$.

These definitions use only ZF. No nonemptiness principle for arbitrary products is being invoked. Continuous-image and Borel-image characterizations require separate proofs; they are not part of this definition. This convention uses the closed-projection characterization in Marker Lemma 4.2(iii), rather than importing the other characterizations from its statement.