Contracting the socle in rings of continuous functions
A short note on separable frames
Following the introduction of separability in frames ([2]) we investigate further properties of this notion and establish some consequences of the Urysohn metrization theorem for frames that are frame counterparts of corresponding results in spaces. In particular we also show that regular subframes of compact metrizable frames are metrizable.
When spectra of lattices of -ideals are Stone-Čech compactifications
Let be a completely regular Hausdorff space and, as usual, let denote the ring of real-valued continuous functions on . The lattice of -ideals of has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) precisely when is a -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a -ideal if whenever two elements have the same annihilator and...
Generalized tri-quotient maps and Čech-completeness
For a topological space let be the set of all compact subsets of . The purpose of this paper is to characterize Lindelöf Čech-complete spaces by means of the sets . Similar characterizations also hold for Lindelöf locally compact , as well as for countably -determined spaces . Our results extend a classical result of J. Christensen.
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