Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.