  Starting from Boolean algebras of languages closed under quotients and using
duality theoretic insights, we derive the notion of Boolean spaces with
internal monoids as recognisers for arbitrary formal languages of finite words
over finite alphabets. This leads to a setting that is well-suited for applying
existing tools from Stone duality as applied in semantics. The main focus of
the paper is the development of topo-algebraic constructions pertinent to the
treatment of languages given by logic formulas. In particular, using the
standard semantic view of quantification as projection, we derive a notion of
Sch\"utzenberger product for Boolean spaces with internal monoids. This makes
heavy use of the Vietoris construction, and its dual functor, which is central
to the coalgebraic treatment of classical modal logic. We show that the unary
Sch\"utzenberger product for spaces, when applied to a recogniser for the
language associated to a formula with a free first-order variable, yields a
recogniser for the language of all models of the corresponding existentially
quantified formula. Further, we generalise global and local versions of the
theorems of Sch\"utzenberger and Reutenauer characterising the languages
recognised by the binary Sch\"utzenberger product. Finally, we provide an
equational characterisation of Boolean algebras obtained by local
Sch\"utzenberger product with the one element space based on an Egli-Milner
type condition on generalised factorisations of ultrafilters on words.
