NOTES ON KURATOWSKI-MRÓWKA THEOREMS - IN POINT-FREE CONTEXT

Sep 4, 2026

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The paper presents Kuratowski-Mrówka compactness characterizations in a point-free setting. It proves that a locale A is compact exactly when the natural projection A×B→B is closed for each locale B. For infinite cardinals α, it establishes an α-compactness criterion restricted to α-discrete spatial locales B, leaving open whether A×B→B is closed for all α-discrete B. It also studies a dual characterization: which B make A×B→B closed for all locales A, relating the result to complete Boolean algebras in regular frames.

LanguageEnglish

CategoryResearch & Report

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NOTES ON KURATOWSKI-MRÓWKA THEOREMS - IN POINT-FREE CONTEXT

Introduction

1. Preliminaries

1.1 Basic conventions

1.2 Frames and locales

1.3 Regularity

1.4 Sublocales

CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLECATÉGORIQUES A.PULTR A.ToZZI Notes on Kuratowski-Mrówka theorems in point-free context Cahiers de topologie et géométrie différentielle catégoriques,tome 33,n1(1992),p.3-14 C Andrée C.Ehresmann et les auteurs,1992,tous droits réservés. L'accès aux archives de la revue《Cahiers de topologie et géométriedifférentielle catégoriques 》implique I'accord avec les conditionsgénérales d'utilisation(http://www.numdam.org/conditions).Touteutilisation commerciale ou impression systématique est constitutived'une infraction pénale.Toute copie ou impression de ce fichierdoit contenir la présente mention de copyright. NUMDAM Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/ # NOTES ON KURATOWsKI-MRÓWKA THEOREMS IN POINT-FREE CONTEXT by A.PULTR¹and A.TOZzI¹ Résumé.Le fameux thóreme de Kuratowski-Mrówka dit qu'un espacetopologique X est compact si et seulement sila projection X×Y→Yest fermée pour tous les espaces Y.Nous démontrons le théoreme deKuratowski dans le domaine des locales.En particulier nous démontronsqu'un locale A est compact si et seulement sila projection A×B一→Best fermee pour tous les locales B.Pour un numbre cardinal infini α lerésultat que nous obtenons n'est pas si satisfaisant.Nous pouvons prouverseulement qu'un localeA est α-compact si et sculcment siA×B一→B estfermee pour tous les locales spatiaux α-discrets. # Introduction The famous Kuratowski theorem characterizes compact spaces X by thefact that for each Y the projection X×Y→Y is closed.Precisely,Ku-ratowski [7]proved that,in the realm of metric spaces,if X is compact thenthe projection πy is closed for any metric space Y,Bourbaki[1]proved thesame property in the category of Hausdorff spaces and Mrówka[9]established theconverse property so that a space X is compact iff the projection πy is closedfor any space Y.Similarly,by results of Noble[10],Vaughan [11]and Giuli[3],aspace X is α-compact([a,β]-compact)iff the projection πy:X×Y→Y isclosed for each α-discrete(α-discrete with character β)space Y.In this paperwe consider these phenomena in the pointfree context. In particular,we prove the Kuratowski theorem in the form that alocale A is compact iff the natural projection A×B→B is closed for eachlocale B.For general α the result we present is not so satisfactory:We are ableto prove only that a locale A is α-compact iff A×B→B is closed for allα-discrete spatial B.Thus,the question whether in this case A×B一→B isclosed for all α-discrete B remains open.Still,the result answers the questionwhether there is a class C(α)of locales such that the α-compactness is character-ized by the closedness of the projections A×B→B with B∈C(α).More- # PULTR &TOZzI-ON KURATOwsKI-MRÓWKA THEOREMS... over,the negative part of the statement presented,namely that for an A whichis not α-compact there is a spatial a-discrete B such that A×B→B is notclosed,is in fact stronger that the respective part of the desired statement Finally,we consider a”dual”of the Kuratowski's characterization,namelythe question as to which B have the property that the projections A×B→Bare closed for all A.In the classical context this characterizes the quasidiscretespaces,hence,in the regular case,the discrete spaces,not a very colourful class.Inthe pointfree context,however,this requirement characterizes the complete Booleanalgebras in among regular frames,which is perharps more interesting. Only basic knowledge of category theory(as e.g.in the introductory chaptersof [8])is assumed.All the necessary facts of pointfree topology are presented inSection 1.For more detail,the reader can consult,e.g.,[5]. ## 1.Preliminaries 1.1.Basic conventions:The cardinality of a set X will be indicatedas |X|.The identity mapping of a set (object)X onto itself will be denotedby idx or simply id.If pi:X₁×X₂→X;is a(categorial)product,t

FAQs

What does the Kuratowski-Mrówka theorem characterize in topology?

It characterizes compact spaces X by the condition that for every space Y the projection X×Y→Y is closed.

How is compactness characterized for a locale in this paper?

A locale A is compact iff the natural projection A×B→B is closed for every locale B.

What is proved in the paper for α-compactness when α is an infinite cardinal?

An α-compact locale A is shown to satisfy that A×B→B is closed for all α-discrete spatial locales B, while the closure property for all α-discrete locales B remains open.

Sep 4, 2026

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