Stably compact spaces

JIMMIE LAWSON
2010 Mathematical Structures in Computer Science  
The purpose of this article is to develop the basic theory of stably compact spaces (=compact, locally compact, coherent sober spaces) and introduce in an accessible manner with a minimum of prerequisites significant new lines of their investigation and application arising from recent research, primarily in the theoretical computer science community. There are three primary themes that are developed: firstly the property of stable compactness is preserved under a large variety of constructions
more » ... nvolving powerdomains, hyperspaces, and function spaces, secondly the underlying de Groot duality of stably compact spaces, which finds varied expression, is reflected by duality theorems involving the just mentioned constructions, and thirdly the notion of inner and outer pavings is a useful and natural tool for such studies of stably compact spaces. Remark 2.3. We note that ↑K, the saturation of K, is compact if K is, and that K ⊆ U implies ↑K ⊆ ↑U = U . Thus we can assume without loss of generality that the compact set K in the definition in (ii) of local compactness is chosen to be saturated, i.e., K = ↑K. A nonempty subset A of a space is irreducible if whenever it is contained in the union of two closed sets, it is contained in one of them. A space X is sober if every closed irreducible set is the closure of a unique singleton set. Remark 2.4. We note that locally compact+well-filtered ⇒ sober ⇒ well-filtered (Gierz et. al. 2003, Theorem II-1.21), so well-filtered (property (iv)) may be replaced by sober in the definition of stably compact. We freely pass between the two equivalent definitions: (i)-(iii)+well-filtered or (i)-(iii)+sober. There are two important categories with objects the stably compact spaces. The first is
doi:10.1017/s0960129510000319 fatcat:uwsasrcwgvasfgr6ruvuhj7squ