Higher order dissolutions and Boolean coreflections of locales

Plewe, Till

doi:10.1016/s0022-4049(99)00193-0

Cited by 29 publications

(10 citation statements)

References 10 publications

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“…It is shown in [13,Lemma 3.2] that h is nowhere dense if and only if h * (0) is a dense element. Further, [11,Proposition 3.9] shows that h: M → L is nowhere dense precisely if, viewed as locales, Fix(h * h) has zero meet (in the co-frame of sublocales) with the smallest dense sublocale of L. Thus, this notion of nowhere density agrees with that of Plewe [30].…”

Section: H(w) ∨ T = 1 and H(w) ∧ T = H(c)mentioning

confidence: 71%

When certain prime ideals in rings of continuous functions are minimal or maximal

Dube

Nsayi

2015

Topology and its Applications

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A Tychonoff space X is called a quasi m-space if every prime d-ideal of the ring C(X) is either a maximal ideal or a minimal prime ideal. These spaces were characterised by Azarpanah and Karavan. In this paper we look at some properties of these spaces from a ring-theoretic perspective. We observe, for instance, that among subspaces which inherit this property are (i) cozero subspaces, (ii) dense z-embedded subspaces, and (iii) regular-closed subspaces among the normal quasi m-spaces. The ring-theoretic approach actually yields the above results within the broader context of frames. The latter part of the paper discusses completely regular frames L for which every prime z-ideal in the ring RL is a maximal ideal or a minimal prime ideal.

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Section: H(w) ∨ T = 1 and H(w) ∧ T = H(c)mentioning

confidence: 71%

When certain prime ideals in rings of continuous functions are minimal or maximal

Dube

Nsayi

2015

Topology and its Applications

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“…Plewe's proof in [18] is in terms of sublocales. For the sake of completeness let us give a proof, but a different one carried out entirely in Frm.…”

Section: Weak Pseudocompactness and Bairenessmentioning

confidence: 99%

A note on weakly pseudocompact locales

Dube

2017

Appl. Gen. Topol.

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We revisit weak pseudocompactness in pointfree topology, and show that a locale is weakly pseudocompact if and only if it is G δ -dense in some compactification. This localic approach (in contrast with the earlier frame-theoretic one) enables us to show that finite localic products of locales whose non-void G δ -sublocales are spatial inherit weak pseudocompactness from the factors. We also show that if a locale is weakly pseudocompact and its G δ -sublocales are complemented then it is Baire.

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“…However, if L is scattered, that is, if every congruence of L is complemented [15] or, equivalently, [16] the conditions in 3.1(4) and 3.1(1) are equivalent, as for spaces: …”

Section: Interior-preserving Coversmentioning

confidence: 99%

“…Further, families {X a | a ∈ A} of open sublocales are distributive [15], that is, satisfy S ∧ a∈A X a = a∈A (S ∧ X a ) for all S ∈ S(X). Every sublocale j : S X has a closure cl(j) : cl(S) X, which is the smallest closed sublocale that contains j and an interior int(j) : int(S) X (the largest open sublocale contained in j).…”

Section: Introductionmentioning

confidence: 99%

On Point-finiteness in Pointfree Topology

Ferreira

Picado

2006

Appl Categor Struct

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In pointfree topology, the point-finite covers introduced by Dowker and Strauss do not behave similarly to their classical counterparts with respect to transitive quasi-uniformities, contrarily to what happens with other familiar types of interior-preserving covers. The purpose of this paper is to remedy this by modifying the definition of Dowker and Strauss. We present arguments to justify that this modification turns out to be the right pointfree definition of point-finiteness. Along the way we place point-finite covers among the classes of interior-preserving and closure-preserving families of covers that are relevant for the theory of (transitive) quasi-uniformities, completing the study initiated with [6].

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Higher order dissolutions and Boolean coreflections of locales

Cited by 29 publications

References 10 publications

When certain prime ideals in rings of continuous functions are minimal or maximal

When certain prime ideals in rings of continuous functions are minimal or maximal

A note on weakly pseudocompact locales

On Point-finiteness in Pointfree Topology

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