Some modifications of Scott's theorem on injective spaces

Abstract

D. Scott in his paper [5] on the mathematical models for the Church-Curry λ-calculus proved the following theorem.

A topological space X. is an absolute extensor for the category of all topological spaces iff a contraction of X. is a topological space of “Scott's open sets” in a continuous lattice.

In this paper we prove a generalization of this theorem for the category of 〈α, δ〉-closure spaces. The main theorem says that, for some cardinal numbers α, δ, absolute extensors for the category of 〈α, δ〉-closure spaces are exactly 〈α, δ〉-closure spaces of 〈α, δ〉-filters in 〈α, δ>-semidistributive lattices (Theorem 3.5).

If α = ω and δ = ∞ we obtain Scott's Theorem (Corollary 2.1). If α = 0 and δ = ω we obtain a characterization of closure spaces of filters in a complete Heyting lattice (Corollary 3.4). If α = 0 and δ = ∞ we obtain a characterization of closure space of all principial filters in a completely distributive complete lattice (Corollary 3.3).