Full intersection types and topologies in lambda calculus

Abstract

Topologies are introduced on the set of lambda terms by their typeability in the full intersection type assignment system. These topologies give rise to simple proofs of some fundamental results of the lambda calculus such as the continuity theorem and the genericity lemma. We show that application is continuous, unsolvable terms are bottoms, and normal forms are isolated points with respect to these topologies. The restriction of all these topologies to the set of closed lambda terms appears to be unique. We compare the introduced topology with the filter topology on the set of (closed) lambda terms and show that they coincide.