scholarly journals Localic remote points revisited

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 111-120 ◽  
Author(s):  
Themba Dube ◽  
Martin Mugochi
Keyword(s):  
Prime Ideals ◽  
Remote Point ◽  
Completely Regular ◽  
Regular Frame ◽  
Completely Regular Frame ◽  
Radical Ideals ◽  
Strict Extension ◽  
Remote Points

We consider remote points in general extensions of frames, with an emphasis on perfect extensions. For a strict extension ?XL ? L determined by a set X of filters in L, we show that if there is an ultrafilter in X then the extension has a remote point. In particular, if a completely regular frame L has a maximal completely regular filter which is an ultrafilter, then ?L ? L has a remote point, where ?L is the Stone-C?ch compactification of L. We prove that in certain extensions associated with radical ideals and l-ideals of reduced f-rings, remote points induced by algebraic data are exactly non-essential prime ideals or non-essential irreducible l-ideals. Concerning coproducts, we show that if M1 ? L1 and M2 ? L2 are extensions of T1-frames, then each of these extensions has a remote point if the extensionM1?M2?L1?L2 has a remote point.

A Pseudocompact Completely Regular Frame which is not Spatial

Order ◽  
2013 ◽  
Vol 31 (1) ◽  
pp. 115-120 ◽  
Author(s):  
Themba Dube ◽  
Stavros Iliadis ◽  
Jan van Mill ◽  
Inderasan Naidoo
Keyword(s):  
Completely Regular ◽  
Regular Frame ◽  
Completely Regular Frame

Compactifications and A-compactifications of frames. Proximal frames

1996 ◽  
Vol 119 (2) ◽  
pp. 321-339
Author(s):  
Georgi D. Dimov ◽  
Gino Tironi
Keyword(s):  
Ordered Set ◽  
Distributive Lattices ◽  
Zero Set ◽  
Completely Regular ◽  
Alexandroff Space ◽  
Regular Frame ◽  
Inclusion Set ◽  
Completely Regular Frame

The aim of this paper is to give two new descriptions of the ordered set of all (up to equivalence) regular compactifications of a completely regular frame. F and to introduce and study the notion of A-frame as a generalization of the notion of Alexandroff space (known also as zero-set space) (Alexandroff[l], Gordon[15]). A description of the ordered set of all (up to equivalence) A-compactifications of an A-frame by means of an ordered by inclusion set of some distributive lattices (called AP-sublattices) is obtained. It implies that any A-frame has a greatest A-compactification and leads to the descriptions of A new category isomorphic to the category of proximal frames is introduced. A question for compactifications of frames analogous to the R. Chandler's question [8, p. 71] for compactifications of spaces is formulated and solved. Many results of [1], [3], [15], [23], [9], [10] and [11] are generalized.

Homeomorphic Sets of Remote Points

1971 ◽  
Vol 23 (3) ◽  
pp. 495-502 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Hausdorff Space ◽  
Continuum Hypothesis ◽  
Locally Compact ◽  
Remote Point ◽  
Completely Regular ◽  
Isolated Points ◽  
Discrete Subspace ◽  
Remote Points ◽  
Separable Metric Space ◽  
The Continuum

Let X be a completely regular Hausdorff space, and let βX denote the Stone-Čech compactification of X. A point p ∈ βX is called a remote point of βX if p does not belong to the βX-closure of any discrete subspace of X. Remote points were first defined and studied by Fine and Gillman, who proved that if the continuum hypothesis is assumed then the set of remote points of βR((βQ) is dense in βR – R(βQ – Q ) (R denotes the space of reals, Q the space of rationals). Assuming the continuum hypothesis, Plank has proved that if X is a locally compact, non-compact, separable metric space without isolated points, then βX has a set of remote points that is dense in βX – X. Robinson has extended this result by dropping the assumption that X is separable.

scholarly journals Strong 0-dimensionality in Pointfree Topology

10.29007/5dmr ◽  
2018 ◽  
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Tychonoff Space ◽  
Completely Regular ◽  
Regular Frame ◽  
Completely Regular Frame

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Cech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen immediately that a Tychonoff space is strongly 0-dimensional iff the frame of its open sets is strongly 0-dimensional in the present sense. This talk will provide an account of various aspects of this notion.

scholarly journals The P-frame reflection of a completely regular frame

2011 ◽  
Vol 158 (14) ◽  
pp. 1778-1794 ◽  
Author(s):  
Richard N. Ball ◽  
Joanne Walters-Wayland ◽  
Eric Zenk
Keyword(s):  
Completely Regular ◽  
Regular Frame ◽  
Frame Reflection ◽  
Completely Regular Frame

scholarly journals REAL IDEALS IN POINTFREE RINGS OF CONTINUOUS FUNCTIONS

2010 ◽  
Vol 83 (2) ◽  
pp. 338-352 ◽  
Author(s):  
THEMBA DUBE
Keyword(s):  
Maximal Ideal ◽  
Classical Case ◽  
Continuous Functions ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Regular Frame ◽  
Completely Regular Frame ◽  
Real Ideals

AbstractReal 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 hyper-real—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.

Nowhere Dense Subsets of Metric Spaces with Applications to Stone-Cech Compactifications

1972 ◽  
Vol 24 (4) ◽  
pp. 622-630 ◽  
Author(s):  
Jack R. Porter ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Hausdorff Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Locally Compact ◽  
Completely Regular ◽  
Isolated Points ◽  
Remote Points ◽  
Topological Boundary ◽  
Regular Closed

Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

Radicals of PID's and Dedekind Domains

1972 ◽  
Vol 24 (4) ◽  
pp. 566-572 ◽  
Author(s):  
R. E. Propes
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Radical Class ◽  
Dedekind Domains ◽  
Radical Ideals ◽  
The One ◽  
Image Position ◽  
Principal Ideal Domains

The purpose of this paper is to characterize the radical ideals of principal ideal domains and Dedekind domains. We show that if T is a radical class and R is a PID, then T(R) is an intersection of prime ideals of R. More specifically, ifthen T(R) = (p1p2 … pk), where p1, p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of R generated by p1p2 … pk. We also characterize the radical ideals of commutative principal ideal rings. For radical ideals of Dedekind domains we obtain a characterization similar to the one given for PID's.

Some Separable Spaces and Remote Points

1982 ◽  
Vol 34 (6) ◽  
pp. 1378-1389 ◽  
Author(s):  
Alan Dow
Keyword(s):  
Dense Subset ◽  
Remote Point ◽  
Separable Spaces ◽  
Remote Points

0. Introduction. A point p ∈ βX\X is called a remote point of X if P ∉ clβXA for each nowhere dense subset A of X. If X is a topological sum Σ{Xn : n ∈ ω} we call nice if {n : F ∩ Xn = ∅} is finite for each . We call remote if for each nowhere dense subset A of X there is an with F ∩ A = ∅ and n-linked if each intersection of at most n elements of is non-empty.

scholarly journals Prime z-Ideals of C(X) and Related Rings

1980 ◽  
Vol 23 (4) ◽  
pp. 437-443 ◽  
Author(s):  
Gordon Mason
Keyword(s):  
Topological Space ◽  
Prime Ideals ◽  
Completely Regular ◽  
The Subject

Let C(X) be the ring of continuous real-valued functions on a (completely regular) topological space X. The structure of the prime ideals and the prime z-ideals of C(X) has been the subject of much investigation (see e-g- [1], [3], [5]). One of the surprising facts about C(X) is that the sum of two prime ideals is again prime.

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