Decomposition Properties of Sober Topologies on the Same Sets 1

In this paper, decomposition properties of sober topologies on the same sets are concerned. Main results are: (1) The supremum of a family of sober topologies on the same set with the same closures on every singletons remains a sober topology; (2) The supremum of a family of T1 and sober topologies on the same set is still a T1 and sober topology; (3) The supremum of a directed family of sober topologies on the same set remains a sober topology. Some subtle (counter)examples are also constructed to show that the supremum of two sober topologies on the same set may not be a sober one.