Localic Metric spaces and the localic Gelfand duality

In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that the constructive Gelfand duality can be extended into a duality between compact regular locales and unital abelian localic C*-algebras. In order to do so we develop a constructive theory of localic metric spaces and localic Banach spaces, we study the notion of localic completion of such objects and the behaviour of these constructions with respect to pull-back along geometric morphisms.Comment: 57 page

On locales whose countably compact sublocales have compact closure

summary:Among completely regular locales, we characterize those that have the feature described in the title. They are, of course, localic analogues of what are called ${\rm cl}$-isocompact spaces. They have been considered in T. Dube, I. Naidoo, C. N. Ncube (2014), so here we give new characterizations that do not appear in this reference

Rings of real functions in pointfree topology

AbstractThis paper deals with the algebra F(L) of real functions on a frame L and its subclasses LSC(L) and USC(L) of, respectively, lower and upper semicontinuous real functions. It is well known that F(L) is a lattice-ordered ring; this paper presents explicit formulas for its algebraic operations which allow to conclude about their behaviour in LSC(L) and USC(L).As applications, idempotent functions are characterized and previous pointfree results about strict insertion of functions are significantly improved: general pointfree formulations that correspond exactly to the classical strict insertion results of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces are derived.The paper ends with a brief discussion concerning the frames in which every arbitrary real function on the α-dissolution of the frame is continuous

A study of localic subspaces, separation, and variants of normality and their duals.

198 p.As in classical topology, in localic topology one often needs to restrict to locales satisfyinga certain degree of separation. In fact, the study of separation in the category of localesconstitutes a non-trivial and important piece of the theory. For instance, it is sometimesimpossible to give an exact counterpart of a classical axiom, while other times a singleproperty for spaces yields multiple non-equivalent localic versions.The main goal of this thesis is to investigate several classes of separated locales and theirconnections with different classes of sublocales, that is, the regular subobjects in the categoryof locales.In particular, we introduce a new diagonal separation and show that it is, in a certainsense, dual to Isbell¿s (strong) Hausdorff property. The duality between suplattices andpreframes, and that between normality and extremal disconnectedness, turn out to be ofspecial interest in this context.Regarding higher separation, we introduce cardinal generalizations of normality andtheir duals (e.g., properties concerning extensions of disjoint families of cozero elements),and give characterizations via suitable insertion or extension results.The lower separation property known as the TD-axiom, also plays an important role inthe thesis. Namely, we investigate the TD-duality between the category of TD-spaces and acertain (non-full) subcategory of the category of locales, identifying the regular subobjects inthe localic side, and provide several applications in point-free topology.Tal como na topologia clássica, também na topologia dos locales (reticulados locais) éfrequente termos que nos restringir a locales que satisfaçam um certo grau de separação.De facto, o estudo de axiomas de separação na categoria dos locales constitui um aspectonão trivial e relevante da teoria. Por exemplo, em alguns casos é impossível termos acontrapartida exacta de um axioma clássico, enquanto noutros casos uma única propriedadepara espaços topológicos produz, na categoria dos locales, diversas versões não equivalentesentre si.O objectivo principal desta tese é investigar várias classes de locales separados e suasconexões com diferentes classes de sublocales (os subobjetos regulares na categoria doslocales).Em particular, introduzimos uma nova propriedade de separação diagonal e mostramosque se trata, em certo sentido, de uma propriedade dual do axioma (forte) de Hausdorffintroduzido por Isbell. As dualidades entre semi-reticulados e reticulados pré-locais, e entrenormalidade e desconexão extrema, acabam por ter um papel relevante neste contexto.Relativamente a axiomas de separação fortes, introduzimos generalizações de normalidade,em função de um cardinal arbitrário, e suas duais (por exemplo, propriedadesenvolvendo extensões de famílias disjuntas de elementos co-zero), e apresentamos caracterizaçõesem termos de propriedades de inserção ou extensão de funções.O axioma TD, uma propriedade de separação muito fraca, também desempenha umpapel importante nesta tese. Especificamente, investigamos a dualidade TD entre a categoriados espaços topológicos TD e uma determinada subcategoria (não plena) da categoria doslocales, identificando os subobjetos regulares na subcategoria de locales, e apresentamosvárias aplicações à topologia sem pontos.Tal y como ocurre en topología clásica, en topología locálica frecuentemente uno tiene querestringir su atención a locales que cumplen cierto grado de separación. De hecho, el estudiode la separación en la categoría de locales es un aspecto no trivial y relevante de la teoría. Enalgunos casos, es imposible dar una contrapartida exacta a un axioma clásico, mientras queen otros casos, una sola propiedad produce multitud de versiones locálicas no equivalentesentre sí.El principal objetivo de esta tesis es investigar varias clases de locales separados y susrelaciones con diferentes clases de sublocales, esto es, los subobjetos regulares en la categoríade locales.En particular, introducimos una nueva separación diagonal, y probamos que es, en ciertosentido, dual al axioma Hausdorff (fuerte) de Isbell. En este contexto, la dualidad entreretículos completos y premarcos, y aquella entre la normalidad y la desconexión extremaresultan ser de especial interés.En cuanto a la separación más fuerte, introducimos generalizaciones cardinales de lanormalidad y sus duales (por ejemplo, propiedades que consisten en la extensión de familiasdisjuntas de elementos cozero), y damos caracterizaciones de las mismas en términos deteoremas de extensión o inserción.Ciertas propiedades de separación más débiles, especialmente el axioma TD, tambiéndesempeñan un papel importante en esta tesis. Específicamente, investigamos la dualidad TDentre la categoría de espacios topológicos TD y cierta subcategoría (no plena) de la categoríade locales, identificando los subobjetos regulares en la categoría de locales, y proporcionamosalgunas aplicaciones en la topología sin puntos

Rings of real functions in Pointfree Topology

This paper deals with the algebra F(L) of real functions of a frame L and its subclasses LSC(L) and USC(L) of, respectively, lower and upper semicontinuous real functions. It is well-known that F(L) is a lattice-ordered ring; this paper presents explicit formulas for its algebraic operations which allow to conclude about their behaviour in LSC(L) and USC(L). As applications, idempotent functions are characterized and the results of [10] about strict insertion of functions are signi cantly improved: general pointfree formulations that correspond exactly to the classical strict insertion results of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces are derived. The paper ends with a brief discussion concerning the frames in which every arbitrary real function on the -dissolution of the frame is continuou

On the Menger and almost Menger properties in locales

[EN] The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober TD-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.The second-named author acknowledges funding from the National Research Foundation of South Africa under Grant 113829.Bayih, T.; Dube, T.; Ighedo, O. (2021). On the Menger and almost Menger properties in locales. Applied General Topology. 22(1):199-221. https://doi.org/10.4995/agt.2021.14915OJS199221221R. N. Ball and J. Walters-Wayland, C- and C*-quotients in pointfree topology, Dissert. Math. (Rozprawy Mat.) 412 (2002), 1-62. https://doi.org/10.4064/dm412-0-1B. Banaschewski and C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 579-589B. Banaschewski and A. Pultr, Variants of openness, Appl. Categ. Structures 2 (1994), 331-350. https://doi.org/10.1007/BF00873038M. Bonanzinga, F. Cammaroto and M. Matveev, Projective versions of selection principles, Topology Appl. 157 (2010), 874-893. https://doi.org/10.1016/j.topol.2009.12.004C. H. Dowker and D. Strauss, Paracompact frames and closed maps, in: Symposia Mathematica, Vol. XVI, pp. 93-116 (Convegno sulla Topologia Insiemistica e Generale, INDAM, Rome, 1973) Academic Press, London, 1975.C. H. Dowker and D. Strauss, Sums in the category of frames, Houston J. Math. 3 (1976), 17-32.T. Dube, M. M. Mugochi and I. Naidoo, Cech completeness in pointfree topology, Quaest. Math. 37 (2014), 49-65. https://doi.org/10.2989/16073606.2013.779986T. Dube, I. Naidoo and C. N. Ncube, Isocompactness in the category of locales, Appl. Categ. Structures 22 (2014), 727-739. https://doi.org/10.1007/s10485-013-9341-8M. J. Ferreira, J. Picado and S. M. Pinto, Remainders in pointfree topology, Topology Appl. 245 (2018), 21-45. https://doi.org/10.1016/j.topol.2018.06.007J. Gutiérrez García, I. Mozo Carollo and J. Picado, Normal semicontinuity and the Dedekind completion of pointfree function rings, Algebra Universalis 75 (2016), 301-330. https://doi.org/10.1007/s00012-016-0378-zW. He and M. Luo, Completely regular proper reflection of locales over a given locale, Proc. Amer. Math. Soc. 141 (2013), 403-408. https://doi.org/10.1090/S0002-9939-2012-11329-2P. T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982.D. Kocev, Menger-type covering properties of topological spaces, Filomat 29 (2015), 99-106. https://doi.org/10.2298/FIL1501099KJ. Madden and J. Vermeer, Lindelöf locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473-480. https://doi.org/10.1017/S0305004100064410J. Picado and A. Pultr, Frames and Locales: topology without points, Frontiers in Mathematics, Springer, Basel, 2012. https://doi.org/10.1007/978-3-0348-0154-6J. Picado and A. Pultr, Axiom $T_D$ and the Simmons sublocale theorem, Comment. Math. Univ. Carolin. 60 (2019), 701-715.J. Picado and A. Pultr, Notes on point-free topology, manuscript.T. Plewe, Sublocale lattices, J. Pure Appl. Algebra 168 (2002), 309-326. https://doi.org/10.1016/S0022-4049(01)00100-1V. Pták, Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41-74. https://doi.org/10.24033/bsmf.1498Y.-K. Song, Some remarks on almost Menger spaces and weakly Menger spaces, Publ. Inst. Math. (Beograd) (N.S.) 112 (2015), 193-198. https://doi.org/10.2298/PIM150513031SJ. J. C. Vermeulen, Proper maps of locales, J. Pure Appl. Algebra 92 (1994), 79-107. https://doi.org/10.1016/0022-4049(94)90047-

Axiom TDT_D and the Simmons sublocale theorem

summary:More precisely, we are analyzing some of H. Simmons, S.\,B. Niefield and K.\,I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_D$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation {between} sublocales and subspaces. S.\,B. Niefield and K.\,I. Rosenthal just mention this axiom in a remark about Simmons' result. In this paper we show that the role of $T_D$ in this question is crucial. Concentration on the properties of $T_D$-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem

Notes on point-free real functions and sublocales

Using the technique of sublocales we present a survey of some known facts (with a few new ones added) on point-free real functions. The subjects treated are, e.g., images and preimages, semicontinuity, algebraic structure (point-free real arithmetics), zero and cozero parts, z-embeddings, z-open and z-closed maps, disconnectivity, small sublocales and supports