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Martin’s Axiom and ω -resolvability of Baire spaces

Fidel Casarrubias-Segura, Fernando Hernández-Hernández, Angel Tamariz-Mascarúa (2010)

Commentationes Mathematicae Universitatis Carolinae

We prove that, assuming MA, every crowded T 0 space X is ω -resolvable if it satisfies one of the following properties: (1) it contains a π -network of cardinality < 𝔠 constituted by infinite sets, (2) χ ( X ) < 𝔠 , (3) X is a T 2 Baire space and c ( X ) 0 and (4) X is a T 1 Baire space and has a network 𝒩 with cardinality < 𝔠 and such that the collection of the finite elements in it constitutes a σ -locally finite family. Furthermore, we prove that the existence of a T 1 Baire irresolvable space is equivalent to the existence of...

Measurable cardinals and category bases

Andrzej Szymański (1991)

Commentationes Mathematicae Universitatis Carolinae

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

Measures of compactness in approach spaces

R. Baekeland, Robert Lowen (1995)

Commentationes Mathematicae Universitatis Carolinae

We investigate whether in the setting of approach spaces there exist measures of relative compactness, (relative) sequential compactness and (relative) countable compactness in the same vein as Kuratowski's measure of compactness. The answer is yes. Not only can we prove that such measures exist, but we can give usable formulas for them and we can prove that they behave nicely with respect to each other in the same way as the classical notions.

Measure-Theoretic Characterizations of Certain Topological Properties

David Buhagiar, Emmanuel Chetcuti, Anatolij Dvurečenskij (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

It is shown that Čech completeness, ultracompleteness and local compactness can be defined by demanding that certain equivalences hold between certain classes of Baire measures or by demanding that certain classes of Baire measures have non-empty support. This shows that these three topological properties are measurable, similarly to the classical examples of compact spaces, pseudo-compact spaces and realcompact spaces.

Menger curvature and Lipschitz parametrizations in metric spaces

Immo Hahlomaa (2005)

Fundamenta Mathematicae

We show that pointwise bounds on the Menger curvature imply Lipschitz parametrization for general compact metric spaces. We also give some estimates on the optimal Lipschitz constants of the parametrizing maps for the metric spaces in Ω(ε), the class of bounded metric spaces E such that the maximum angle for every triple in E is at least π/2 + arcsinε. Finally, we extend Peter Jones's travelling salesman theorem to general metric spaces.

Metric enrichment, finite generation, and the path coreflection

Alexandru Chirvasitu (2024)

Archivum Mathematicum

We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally 1 -presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry- 0 -generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results include...

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