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On the commutativity of the powerspace constructions. (English) Zbl 07093560
Summary: We investigate powerspace constructions on topological spaces, with a particular focus on the category of quasi-Polish spaces. We show that the upper and lower powerspaces commute on all quasi-Polish spaces, and show more generally that this commutativity is equivalent to the topological property of consonance. We then investigate powerspace constructions on the open set lattices of quasi-Polish spaces, and provide a complete characterization of how the upper and lower powerspaces distribute over the open set lattice construction.

MSC:
06D22 Frames, locales
03E15 Descriptive set theory
06B35 Continuous lattices and posets, applications
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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