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Borel extensions of Baire measures

J. Aldaz (1997)

Fundamenta Mathematicae

We show that in a countably metacompact space, if a Baire measure admits a Borel extension, then it admits a regular Borel extension. We also prove that under the special axiom ♣ there is a Dowker space which is quasi-Mařík but not Mařík, answering a question of H. Ohta and K. Tamano, and under P(c), that there is a Mařík Dowker space, answering a question of W. Adamski. We answer further questions of H. Ohta and K. Tamano by showing that the union of a Mařík space and a compact space is Mařík,...

Borel extensions of Baire measures in ZFC

Menachem Kojman, Henryk Michalewski (2011)

Fundamenta Mathematicae

We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

Calibres, compacta and diagonals

Paul Gartside, Jeremiah Morgan (2016)

Fundamenta Mathematicae

For a space Z let 𝒦(Z) denote the partially ordered set of all compact subspaces of Z under set inclusion. If X is a compact space, Δ is the diagonal in X², and 𝒦(X²∖Δ) has calibre (ω₁,ω), then X is metrizable. There is a compact space X such that X²∖Δ has relative calibre (ω₁,ω) in 𝒦(X²∖Δ), but which is not metrizable. Questions of Cascales et al. (2011) concerning order constraints on 𝒦(A) for every subspace of a space X are answered.

Centered-Lindelöfness versus star-Lindelöfness

Maddalena Bonanzinga, Mikhail Valerʹevich Matveev (2000)

Commentationes Mathematicae Universitatis Carolinae

We discuss various generalizations of the class of Lindelöf spaces and study the difference between two of these generalizations, the classes of star-Lindelöf and centered-Lindelöf spaces.

Complete ℵ 0 -bounded groups need not be ℝ -factorizable

Mihail G. Tkachenko (2001)

Commentationes Mathematicae Universitatis Carolinae

We present an example of a complete ℵ 0 -bounded topological group H which is not ℝ -factorizable. In addition, every G δ -set in the group H is open, but H is not Lindelöf.

Completely regular spaces

H. L. Bentley, Eva Lowen-Colebunders (1991)

Commentationes Mathematicae Universitatis Carolinae

We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the divergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy...

Connected economically metrizable spaces

Taras Banakh, Myroslava Vovk, Michał Ryszard Wójcik (2011)

Fundamenta Mathematicae

A topological space is non-separably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a non-separably connected complete metric space X under a monotone quotient map. The metric d X of the space X is economical in the sense that for each infinite subspace A ⊂ X the cardinality of the set d X ( a , b ) : a , b ∈ A does not exceed the density of A, | d X ( A × A ) | ≤ d e n s ( A ) . The construction of the space X determines a functor : Top...

Countably metacompact spaces in the constructible universe

Paul Szeptycki (1993)

Fundamenta Mathematicae

We present a construction from ♢* of a first countable, regular, countably metacompact space with a closed discrete subspace that is not a G δ . In addition some nonperfect spaces with σ-disjoint bases are constructed.

Disconnectedness properties of hyperspaces

Rodrigo Hernández-Gutiérrez, Angel Tamariz-Mascarúa (2011)

Commentationes Mathematicae Universitatis Carolinae

Let X be a Hausdorff space and let ℋ be one of the hyperspaces C L ( X ) , 𝒦 ( X ) , ℱ ( X ) or ℱ n ( X ) ( n a positive integer) with the Vietoris topology. We study the following disconnectedness properties for ℋ : extremal disconnectedness, being a F ' -space, P -space or weak P -space and hereditary disconnectedness. Our main result states: if X is Hausdorff and F ⊂ X is a closed subset such that (a) both F and X - F are totally disconnected, (b) the quotient X / F is hereditarily disconnected, then 𝒦 ( X ) is hereditarily disconnected. We also...

Discrete homotopy theory and critical values of metric spaces

Jim Conant, Victoria Curnutte, Corey Jones, Conrad Plaut, Kristen Pueschel, Maria Lusby, Jay Wilkins (2014)

Fundamenta Mathematicae

Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3/2. The latter two spectra are known to be discrete for compact geodesic spaces,...

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