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When spectra of lattices of z -ideals are Stone-Čech compactifications

Themba Dube (2017)

Mathematica Bohemica

Let X be a completely regular Hausdorff space and, as usual, let C ( X ) denote the ring of real-valued continuous functions on X . The lattice of z -ideals of C ( X ) has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) β X precisely when X is a P -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 d -ideal if whenever two elements have the same annihilator and...

Which topological spaces have a weak reflection in compact spaces?

Martin Maria Kovár (1995)

Commentationes Mathematicae Universitatis Carolinae

The problem, whether every topological space has a weak compact reflection, was answered by M. Hušek in the negative. Assuming normality, M. Hušek fully characterized the spaces having a weak reflection in compact spaces as the spaces with the finite Wallman remainder. In this paper we prove that the assumption of normality may be omitted. On the other hand, we show that some covering properties kill the weak reflectivity of a noncompact topological space in compact spaces.

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