  We propose a novel topological perspective on data languages recognizable by
orbit-finite nominal monoids. For this purpose, we introduce pro-orbit-finite
nominal topological spaces. Assuming globally bounded support sizes, they
coincide with nominal Stone spaces and are shown to be dually equivalent to a
subcategory of nominal boolean algebras. Recognizable data languages are
characterized as topologically clopen sets of pro-orbit-finite words. In
addition, we explore the expressive power of pro-orbit-finite equations by
establishing a nominal version of Reiterman's pseudovariety theorem.
