If \(X\) is a set, \(X^*\) denotes a nonstandard extension of \(X\). A client space is a pair \((X, \mu)\) where \(X\) is a set and \(\mu= \{\mu (x)\mid x\in X\}\) is a collection of subsets \(\mu (x)\subseteq X^*\), with \(x\in \mu(x)\). A map between client spaces is a function \(f: X\to Y\) with \(f^* (\mu (x)) \subseteq \mu(f (x))\). The category of topological spaces is a reflective subcategory of the category of client spaces. It is shown that the category of client spaces is Cartesian closed and, in fact, essentially forms a topological universe in the sense of L. D. Nel [Contemp. Math. 30, 244-276 (1984; Zbl 0548.46054)]. In addition, the client space structure on the function space \(Y^X\) is shown to agree with well-known function space structures when \(X\) and \(Y\) have appropriate topological structures.