Induced Topologies on the Poset of Finitely Generated Saturated Sets

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Abstract

In [R. Heckmann, K. Keimel, Quasicontinuous Domains and the Smyth Powerdomain, Electronic Notes in Theoretical Computer Science 298 (2013), 215–232], Heckmann and Keimel proved that a dcpo P is quasicontinuous iff the poset Fin P of nonempty finitely generated upper sets ordered by reverse inclusion is continuous. We generalize this result to general topological spaces in this paper. More precisely, for any T0 space (X,τ) and Uτ, we construct a topology τF generated by the basic open subsets UF={FFinX:FU}. It is shown that a T0 space (X,τ) is a hypercontinuous lattice iff τF is a completely distributive lattice. In particular, we prove that if a poset P satisfies property DINTop, then P is quasi-hypercontinuous iff Fin P is hypercontinuous.

Keywords

Hypercontinuous poset
quasicontinuous domain
Scott topology
upper topology

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1

Supported by the National Natural Science Foundation of China (Nos. 11661057, 11701238, 11626121) and the Natural Science Foundation of Jiangxi Province (Nos. 20161BAB2061004, 20161BAB211017)

2

Corresponding author.