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 SC-posets 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, LC-continuity, 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.