Awodey, Steven Topological Representation of the Lambda-Calculus The [lambda]-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of [lambda]-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is [lambda]-definable. These results subsume earlier ones using cartesian closed categories, as well as those employing so-called Henkin and Kripke [lambda]-models. Philosophy 1999-03-05
    https://kilthub.cmu.edu/articles/journal_contribution/Topological_Representation_of_the_Lambda-Calculus/6492968
10.1184/R1/6492968.v1