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.

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.

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.

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.

scholarly journals A NOTE ON RELATIVE PSEUDOCOMPACTNESS IN THE CATEGORY OF FRAMES

2012 ◽  
Vol 87 (1) ◽  
pp. 120-130 ◽  
Author(s):  
THEMBA DUBE
Keyword(s):  
Tychonoff Space ◽  
Completely Regular ◽  
Tychonoff Spaces

AbstractA subspace S of Tychonoff space X is relatively pseudocompact in X if every f∈C(X) is bounded on S. As is well known, this property is characterisable in terms of the functor υ which reflects Tychonoff spaces onto the realcompact ones. A device which exists in the category CRegFrm of completely regular frames which has no counterpart in Tych is the functor which coreflects completely regular frames onto the Lindelöf ones. In this paper we use this functor to characterise relative pseudocompactness.

scholarly journals Characterized Fuzzy R2.5 and Characterized Fuzzy T3.5 Spaces

2017 ◽  
Vol 13 (1) ◽  
pp. 7048-7073
Author(s):  
Ahmed Saeed Abd-Allah
Keyword(s):  
Topological Space ◽  
Tychonoff Space ◽  
Proximity Space ◽  
Completely Regular ◽  
Fuzzy Function ◽  
Continuous Mappings ◽  
Fuzzy Space ◽  
Fuzzy Topological Space ◽  
The Given ◽  
Proximity Spaces

This paper, deals with, introduce and study the notions of haracterized fuzzy R2.5 spaces and of characterized fuzzy T3.5 spaces by using the notion of fuzzy function family presented in [21] and the notion of φ1,2ψ1,2-fuzzy continuous mappings presented in [5] as a generalization of all the weaker and stronger forms of the fuzzy completely regular spaces introduced in [11,24,26,29]. We denote by characterized fuzzy T3.5 space or characterized fuzzy Tychonoff space to the characterized fuzzy space which is characterized fuzzy T1 and characterized fuzzy R2.5 space in this sense. The relations between the characterized fuzzy T3.5 spaces, the characterized fuzzy T4 spaces and the characterized fuzzy T3 spaces are introduced. When the given fuzzy topological space is normal, then the related characterized fuzzy space is finer than the associated characterized fuzzy proximity space which is presented in [1]. Moreover, the associated characterized fuzzy proximity spaces and the characterized fuzzy T4 spaces are identical with help of the complementarilysymmetric fuzzy topogenous structure, that is, identified with the fuzzy proximity δ. More generally, the fuzzy function family of all φ1,2ψ1,2-fuzzy continuous mappings are applied to show that the characterized fuzzy R2.5 spaces and the associated characterized fuzzy proximity spaces are identical.

scholarly journals On the lattice of varieties of completely regular semigroups

1983 ◽  
Vol 35 (2) ◽  
pp. 227-235 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Variety Of Groups ◽  
Completely Regular Semigroups ◽  
Subdirect Products

AbstractSeveral morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

scholarly journals Complete regularity of Ellis semigroups of -actions

2020 ◽  
pp. 1-17
Author(s):  
MARCY BARGE ◽  
JOHANNES KELLENDONK
Keyword(s):  
Complete Regularity ◽  
Completely Regular ◽  
Ellis Semigroup ◽  
Totally Disconnected

Abstract It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

scholarly journals Bounded resolutions for spaces $$C_{p}(X)$$ and a characterization in terms of X

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
J. C. Ferrando ◽  
J. Ka̧kol ◽  
W. Śliwa
Keyword(s):  
Tychonoff Space ◽  
Cosmic Space ◽  
Tychonoff Spaces

AbstractAn internal characterization of the Arkhangel’skiĭ-Calbrix main theorem from [4] is obtained by showing that the space $$C_{p}(X)$$ C p ( X ) of continuous real-valued functions on a Tychonoff space X is K-analytic framed in $$\mathbb {R}^{X}$$ R X if and only if X admits a nice framing. This applies to show that a metrizable (or cosmic) space X is $$\sigma $$ σ -compact if and only if X has a nice framing. We analyse a few concepts which are useful while studying nice framings. For example, a class of Tychonoff spaces X containing strictly Lindelöf Čech-complete spaces is introduced for which a variant of Arkhangel’skiĭ-Calbrix theorem for $$\sigma $$ σ -boundedness of X is shown.

Sign in / Sign up

Export Citation Format

Share Document