Stone-Čech compactification of locales II

As discovered recently, Li and Wang's 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), [271][272][273][274][275][276][277][278][279][280][281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem.

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“…and [2]). The frame of reals is the frame L(R) generated by all ordered pairs (p, q), where p, q ∈ Q, subject to the following relations:…”

Section: Semicontinuity Of Real Functions On Framesmentioning

confidence: 99%

Lower and upper regularizations of frame semicontinuous real functions

2009

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“…Compare this with the Stone-Čech compactification as constructed in [3]. There, one considers the completely regular ideals J in the sense that for a ∈ J there is a b ∈ J such that a ≺ ≺ b.…”

Section: Samuel Compactificationmentioning

confidence: 99%

“…Recall the Stone-Čech compactification from [3]. There, for a completely regular frame L, one took the frame KL of completely regular ideals on L; it turns out that the map v L = (J → J) : KL → L is a dense onto frame homomorphism, with right adjoint…”

Section: An Alternative Description Of Stone-čech Compactificationmentioning

confidence: 99%

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Completion and Samuel compactification of nearness and uniform frames

Banaschewski¹,

Pultr²

2012

Port. Math.

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Abstract. It is shown that the familiar description of the completion of a uniform frame in terms of its Samuel compactification can be extended to arbitrary nearness frames. This is achieved by means of the following new notion, a variant of compactness, for regular frames: such a frame will be called near-compact if it is complete in some totally bounded nearness. This leads to a natural concept of the Samuel near-compactification for arbitrary nearness frames which is then shown to play exactly the same rôle in the general setting which the Samuel compactification plays for uniform frames.

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“…To give a few examples: first, there is a constructive "Stone-Cech compactification" for locales [3], which specializes in the presence of the Prime Ideal Theorem to the usual one for spaces. The minimal projective cover of a space, first constructed by Gleason [18] for (locally) compact Hausdorff spaces and subsequently extended by other authors to more general classes of spaces, requires the Prime Ideal Theorem since it is defined as a space of prime filters; but it has a localic version [31] which works without any such assistance (and which, moreover, is as simple to construct for arbitrary locales as for compact regular ones, in constrast to what happens for spaces).…”

Section: Compact Hausdorff Spaces At This Point a Topologist (Echoimentioning

confidence: 99%