## Sober spaces and sober sets.(English)Zbl 1151.06010

In this article a quasi-ordered set $$(X,\leq)$$ is called quasi-sober if there exists a quasi-sober topology on $$X$$ such that $$\overline{\{x\}}= ({x\uparrow})$$ whenever $$x\in X.$$ Moreover if $$\leq$$ is an ordering and $$(X,\leq)$$ is quasi-sober, then $$(X,\leq)$$ is said to be a sober set. The authors are interested in order-theoretic characterizations of sober sets and, among other things, discuss connections with the theory of spectral posets.

### MSC:

 06F30 Ordered topological structures 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces

### Keywords:

sober set; sober space; spectral set; quasi-sober