A
Suslin scheme is a family P = \{ P_s: s \in \omega^{ of subsets of a set X indexed by finite sequences of non-negative integers. The
Suslin operation applied to this scheme produces the set :\mathcal A P = \bigcup_{x \in {\omega ^ \omega}} \bigcap_{n \in \omega} P_{x \upharpoonright n} Alternatively, suppose we have a
Suslin scheme, in other words a function M from finite sequences of positive integers n_1,\dots, n_k to sets M_{n_1,..., n_k}. The result of the Suslin operation is the set : \mathcal A(M) = \bigcup \left(M_{n_1} \cap M_{n_1, n_2} \cap M_{n_1, n_2, n_3} \cap \dots \right) where the union is taken over all infinite sequences n_1,\dots, n_k, \dots If M is a family of subsets of a set X, then \mathcal A(M) is the family of subsets of X obtained by applying the Suslin operation \mathcal A to all collections as above where all the sets M_{n_1,..., n_k} are in M. The Suslin operation on collections of subsets of X has the property that \mathcal A(\mathcal A(M)) = \mathcal A(M). The family \mathcal A(M) is closed under taking countable intersections andโif X\in Mโcountable unions, but is not in general closed under taking complements. If M is the family of
closed subsets of a
topological space, then the elements of \mathcal A(M) are called
Suslin sets, or
analytic sets if the space is a
Polish space. ==Example==