It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes.

A short elementary proof is given in Srivastava "A course on Borel sets", Theorem 2.6.7. This proof does not proceed through a Borel or analytic cross section for the equivalence relation. Is it possible that a closed equivalence relation can have no analytic cross section?

(A "cross section", or "transversal", is a set with exactly one point in each equivalence class. An equivalence relation on a space $X$ is "closed" if it is a closed subset of the product space $X\times X$.)

Several results asserting the existence of a Borel or analytic cross section assume additional conditions, such as requiring the equivalence classes to be closed, for example.

The stated theorem is a particular instance of a much deeper theorem of J. Silver: every coanalytic equivalence relation has either countably many or $\mathfrak{c}$ many equivalence classes. For a proof, see Jech "Set Theory", 3rd ed. 2002/2006, Chapter 32.