A footnote on local compactness

A standard example of an impredicative definition is that of the way-below relation [11], and thus of local compactness in (point-free) topology [6]. Local compactness can be dealt with predicatively as in [8], [5]; the particularly convenient re-formulation to be presented is based on a recent result by Peter Aczel [2], and on the observation that, in set-presented topologies, the way-below relation can be formulated predicatively (Proposition 3). A useful simple result (Lemma 2), exploited in the predicative formulation of the full way-below relation, is also used to show that a topology isomorphic to a set-presented topology is set-presented (Theorem 5).