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A group topology on the free abelian group of cardinality 𝔠 that makes its square countably compact

Ana Carolina Boero, Artur Hideyuki Tomita (2011)

Fundamenta Mathematicae

Under 𝔭 = 𝔠, we prove that it is possible to endow the free abelian group of cardinality 𝔠 with a group topology that makes its square countably compact. This answers a question posed by Madariaga-Garcia and Tomita and by Tkachenko. We also prove that there exists a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group) whose square is countably compact. This answers a question posed by Grant.

A minimal regular ring extension of C(X)

M. Henriksen, R. Raphael, R. G. Woods (2002)

Fundamenta Mathematicae

Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring C ( X , τ δ ) of continuous real-valued functions on the space ( X , τ δ ) , where τ δ is the smallest Tikhonov topology on X for which τ ⊆ τ δ and C ( X , τ δ ) is von Neumann regular. The compact and metric spaces for which G ( X ) = C ( X , τ δ ) are characterized. Necessary, and different sufficient, conditions...

A nice subclass of functionally countable spaces

Vladimir Vladimirovich Tkachuk (2018)

Commentationes Mathematicae Universitatis Carolinae

A space X is functionally countable if f ( X ) is countable for any continuous function f : X → ℝ . We will call a space X exponentially separable if for any countable family ℱ of closed subsets of X , there exists a countable set A ⊂ X such that A ∩ ⋂ 𝒢 ≠ ∅ whenever 𝒢 ⊂ ℱ and ⋂ 𝒢 ≠ ∅ . Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has...

A Note on Differentiability of Lipschitz Maps

Rafał Górak (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that every Lipschitz map defined on an open subset of the Banach space C(K), where K is a scattered compactum, with values in a Banach space with the Radon-Nikodym property, has a point of Fréchet differentiability. This is a strengthening of the result of Lindenstrauss and Preiss who proved that for countable compacta. As a consequence of the above and a result of Arvanitakis we prove that Lipschitz functions on certain function spaces are Gâteaux differentiable.

A note on transitively D -spaces

Liang-Xue Peng (2011)

Czechoslovak Mathematical Journal

In this note, we show that if for any transitive neighborhood assignment φ for X there is a point-countable refinement ℱ such that for any non-closed subset A of X there is some V ∈ ℱ such that | V ∩ A | ≥ ω , then X is transitively D . As a corollary, if X is a sequential space and has a point-countable w c s * -network then X is transitively D , and hence if X is a Hausdorff k -space and has a point-countable k -network, then X is transitively D . We prove that if X is a countably compact sequential space and has a point-countable...

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