Some modifications of Scott's theorem on injective spaces

✍ Scribed by Andrzej W. Jankowski

• Publisher: Springer Netherlands
• Year: 1986
• Tongue: English
• Weight: 507 KB
• Volume: 45
• Category: Article
• ISSN: 0039-3215
• DOI: 10.1007/bf00373271

✦ Synopsis

D. Scott in his paper [5] on the mathematical models for the Church-Curry 2-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 conti. nuous lattice.
In this paper we prove a generalization of this theorem for the category of closure spaces. The main theorem says that, for some cardinal numbers a, ~, abso. lute extensors for the category of -closure spaces are exactly -closure spaces of -fflters in -semidistributive lattices (Theorem 3.5).
If a = co and (~ = oo we obtain Scott's Theorem (Corollary 2.1). If a = 0 and = co we obtain a characterization of closure spaces of filters in a complete Heyting lattice (Corollary 3.4). If a = 0 and ~ = oo we obtain a characterization of closure space of all principial filters in a completely distributive complete lattice (Corollary
3.3).