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Complete Congruences on Topologies and Down-set Lattices

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Abstract

From the work of Simmons about nuclei in frames it follows that a topological space X is scattered if and only if each congruence Θ on the frame of open sets is induced by a unique subspace A so that \(\Theta = \{ (U,V) | U\mathop{\cap} A = V\mathop{\cap} A\}\), and that the same holds without the uniqueness requirement iff X is weakly scattered (corrupt). We prove a seemingly similar but substantially different result about quasidiscrete topologies (in which arbitrary intersections of open sets are open): each complete congruence on such a topology is induced by a subspace if and only if the corresponding poset is (order) scattered, i.e. contains no dense chain. More questions concerning relations between frame, complete, spatial, induced and open congruences are discussed as well.

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Correspondence to Aleš Pultr.

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Dedicated to Bernhard Banaschewski.

The second author was partially supported by NSF grant no. NSF01-4- 21760, and both the second and the third author would like to express their thanks for support by the Carlsberg Foundation. The third author would like to express his thanks for support by project 1M0021620808 of the Ministry of Education of the Czech Republic.

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Erné, M., Gehrke, M. & Pultr, A. Complete Congruences on Topologies and Down-set Lattices. Appl Categor Struct 15, 163–184 (2007). https://doi.org/10.1007/s10485-006-9054-3

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