Topological representation of the lambda-calculus S. Awodey Carnegie Mellon University September 1998 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.