Lindelöf locales and realcompactness

Madden, James J.; Vermeer, Johannes

doi:10.1017/s0305004100064410

Cited by 76 publications

(38 citation statements)

References 9 publications

(14 reference statements)

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“…In [9] it is shown that H and Coz induce an equivalence between the categories LRegFrm (regular Lindelöf frames) and RegσFrm.…”

Section: Preliminariesmentioning

confidence: 99%

On completion in the category SSNσFrm

Naidoo

2013

Mathematica Slovaca

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ABSTRACT. We introduce the category SSNσFrm of super strong nearness σ-frames and show the existence of a completion for a super strong nearness σ-frame unique up to isomorphism by the similar construction presented in Completion is also shown to be a coreflection in SSNσFrm. We also engage with the notion of total boundedness for nearness σ-frames and provide a characterization of the Samuel compactification of a nearness σ-frame alternative to the description in [NAIDOO, I.: Samuel compactification and uniform coreflection of nearness σ-frames, Czechoslovak Math.

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“…In [9] it is shown that H and Coz induce an equivalence between the categories LRegFrm (regular Lindelöf frames) and RegσFrm.…”

Section: Preliminariesmentioning

confidence: 99%

On completion in the category SSNσFrm

Naidoo

2013

Mathematica Slovaca

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“…Using localic language, Madden and Vermeer [15] have shown that regular Lindelöf locales form a reflective subcategory of the category of locales by actually constructing the reflection, λL, for any completely regular locale L. We recall the construction in frame terms because that is the category of discourse in this paper.…”

Section: The Coreflections βL λL and υLmentioning

confidence: 99%

A broader view of the almost Lindelöf property

Dube

2011

Algebra Univers.

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By first finding necessary and sufficient conditions for the realcompact coreflection, υL, and the regular Lindelöf coreflection, λL, of a completely regular frame L to be isomorphic, we define a frame L to be almost Lindelöf if it is Lindelöf or λL → L is a one-point extension. This agrees with the condition "υL is Lindelöf and L is realcompact or υL is a one-point extension", which would be a frame version of what are called almost Lindelöf spaces. Thus, the condition "υX is Lindelöf", which is added in the definition of almost Lindelöf spaces, serves only to compensate for the lack of the regular Lindelöf reflection in Top, and can be dispensed with by concentrating on the frame OX instead of the space X.

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“…The regular Lindelöf coreflection of L, denoted λL, is the frame of σ -ideals of Coz L (see [15]). The frame υ L is constructed in the following manner.…”

Section: Real and Hyper-real Ideals Of Rlmentioning

confidence: 99%

Real Ideals in Pointfree Rings of Continuous Functions

Dube

2010

Bull. Aust. Math. Soc.

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Real ideals of the ring RL of real-valued continuous functions on a completely regular frame L are characterized in terms of cozero elements, in the manner of the classical case of the rings C(X ). As an application, we show that L is realcompact if and only if every free maximal ideal of RL is hyperreal-which is the precise translation of how Hewitt defined realcompact spaces, albeit under a different appellation. We also obtain a frame version of Mrówka's theorem that characterizes realcompact spaces.2010 Mathematics subject classification: primary 06D22; secondary 46E25, 16D25.

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Lindelöf locales and realcompactness

Cited by 76 publications

References 9 publications

On completion in the category SSNσFrm

On completion in the category SSNσFrm

A broader view of the almost Lindelöf property

Real Ideals in Pointfree Rings of Continuous Functions

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