Lifting Of Sobriety Concepts With Particular Reference To (L, M)-Topological Spaces

Abstract

The concept of sobriety of ordinary topological spaces has been around since at least the 1970’s. Sober topological spaces were introduced by Grothendieck et al ([1] IV 4.2.1) and independently by T. Blanksma [3]. Since then various authors studied this axiom, e.g. R-E. Hoffmann ([5] and [6]) and P. T. Johnstone [10]. It was pointed out by [21] and [25] that this axiom is also of importance to computer science, domain theory in particular. Smyth argues that “computationally reasonable spaces are sober”. There are several reasons for this, one being the possibility of, in the case of sobriety, moving from frame maps between frames to continuous maps between topological spaces; in other words, continuous maps being categorically (dually) equivalent to frame maps requires sobriety (see e.g. [11], Theorems 3.3 and 3.4). Another is that the Scott topology on continuous posets are highly non-Hausdorff but sober. See e.g. [10]. (The Scott topology on a two point set is the Sierpinski space).