Strongly continuous posets and the local Scott topology ✩

In this paper, the concept of strongly continuous posets (SC-posets, for short) is introduced. A new intrinsic topology—the local Scott topology is defined and used to characterize SCposets and weak monotone convergence spaces. Four notions of continuity on posets are compared in detail and some subtle counterexamples are constructed. Main results are: (1) A poset is an SC-poset iff its local Scott topology is equal to its Scott topology and is completely distributive iff it is a continuous precup; (2) For precups, PI-continuity, LCcontinuity, SC-continuity and the usual continuity are equal, whereas they are mutually different for general posets; (3) A T0-space is an SC-poset equipped with the Scott topology iff the space is a weak monotone convergence space with a completely distributive topology contained in the local Scott topology of the specialization order