Posets determined by countably many core-compact subspaces

Let me return to a topic that I have addressed a few times in recent posts: under which conditions does the Scott topology of the product of two posets coincide with the product topology of the Scott topologies? We have seen quite a few cases already, and this month we will see that this holds for cc_ω-posets, namely posets whose Scott topology is determined by the Scott topology of a countably ascending sequence of sub-d-posets, each of them being core-compact in its Scott topology. The construction is due to Lawson and Xu. The cc_ω-posets are exactly certain forms of colimits of posets in the category Pos_σ of posets and Scott-continuous maps, and the whole point is that not only are those colimits also colimits in Top, but those form colimits in Top over which products distribute (see the post of previous month). Read the full post.