On points of convergence lattices and sobriety for convergence spaces
F. Mynard
TL;DR
The paper extends sobriety and sobrification concepts from topological spaces to general convergence spaces by studying point constructions pt L in convergence lattices and their quotients. It develops a detailed framework connecting irreducible filters, T_D, and various sobriety notions, and shows how the space of points pt((P X, lim_ξ)) relates to the original X via dense embeddings, with a key quotient pt′L recovering sobrification in the topological case. The authors introduce Z-regularity and analyze how open/closed elements govern the topological behavior of pt L and pt′L, establishing when sobrification can be captured in the pointfree setting. They propose Problem ['prob:main'] to characterize when irreducible filters collapse to principal ones under weak sobriety and T_D, linking these conditions to the possibility that pt_L is homeomorphic to X. Overall, the work provides a cohesive pointfree perspective on sobriety, T_D, and sobrification that unifies and extends classical results from topology to convergence spaces.
Abstract
We characterize the convergence spaces $(X,ξ)$ such that the space of points of $(\mathbb{P}X,\lim_ξ)$ in the category of convergence lattices is $(X,ξ)$. On the way, we study variants of sobriety and of the axiom $T_{D}$ in convergence spaces. New phenomena appear when leaving the realm of topological spaces. We obtain new hindsight into the space of points of a convergence lattice and study a special quotient of it, which, in the case $L=(\mathbb{P}X,\lim_ξ)$ for a topological space $(X,ξ)$, turns out to be homeomorphic to the sobrification of $X$.