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Realcompact spaces and regular \(\sigma\) -frames. (English) Zbl 0547.54021
A lattice is called a regular \(\sigma\) -frame if it has countable joins, top e, bottom O, satisfies \(x\wedge\bigvee^{\infty}_{n=1}x_ n=\bigvee^{\infty}_{n=1}(x\wedge x_ n)\), and each element a is a countable join of elements x rather below it, i.e. \(s\vee a=e\) for some element s disjoint from x. A connection (in the category-theoretic sense) is established with the category of zero-set spaces (called Alexandroff spaces). Various aspects of this connection are studied, in particular the objects fixed by the unit and the co-unit of the connection. Thus a duality is established between the category of realcompact zero-set spaces and that of regular \(\sigma\) -frames with enough \(\sigma\) -prime filters to separate elements.
Reviewer: H.Eugenes

54D60 Realcompactness and realcompactification
06B10 Lattice ideals, congruence relations
18B30 Categories of topological spaces and continuous mappings (MSC2010)
06B15 Representation theory of lattices
Full Text: DOI
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