On products of dcpos, the Miao-Xi-Li-Zhao lemma, and the complete lattice Lfan (part I)

The product of a poset P with itself can be given two topologies: the Scott topology of the product, or the product of the Scott topologies. Those two topologies differ in general, but they coincide when P is a continuous poset, and more generally when P is core-compact, or first-countable. We will see a (very) simple example of a dcpo—even a complete lattice—for which those two topologies coincide, but that is neither core-compact nor first-countable. This was discovered independently by Xu, Shen, Xi and Zhao in 2020, by Chen, Kou and Lyu in 2022 and by Hertling in 2022. Showing that the Scott topology of the product coincides with the product of the Scott topologies is best done through a lemma due to Miao, Xi, Li, and Zhao in 2022, and which says the curious thing that this coincidence of topologies happens as soon as there are only countably many ideals that have suprema on the given poset. Read the full post.