What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

Solution 1:

This is taken from T. Jech's "Set Theory" book Ch. Lemma 11.6 provides equivalent definitions of analytic sets in a polish space X, most of them require projection of a product of 2 spaces (Baire space and X). In order to "stay within the boundries" of the space X, you need the Suslin operation A. At the end of the chapter Jech mentions that "Suslin’s discovery of an error in a proof in Lebesgue’s article led to a construction of an analytic non-Borel set and introduction of the operation A."

Alternatively if you take the recursive definition of the analytic sets as $\Sigma^1_1(x)$ sets (x - a real) then simplifying the recursive function/formula (Turing machine) defined by a given $\Sigma^1_1(x)$ set you realize why adding a second order quantifier ($\Sigma^1$) is equal to a projection of a closed sets. For the recursive definition of the projective hierarchy see D. Marker Descriptive Set Theory.