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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 hit-and-miss topology for 2 X , Cₙ(X) and Fₙ(X)

Benjamín Espinoza, Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano (2009)

Colloquium Mathematicae

A hit-and-miss topology ( τ H M ) is defined for the hyperspaces 2 X , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between τ H M and the Vietoris topology and we find conditions on X for which these topologies are equivalent.

A new approach for KM-fuzzy partial metric spaces

Yu Shen, Chong Shen, Conghua Yan (2022)

Kybernetika

The main purpose of this paper is to give a new approach for partial metric spaces. We first provide the new concept of KM-fuzzy partial metric, as an extension of both the partial metric and KM-fuzzy metric. Then its relationship with the KM-fuzzy quasi-metric is established. In particularly, we construct a KM-fuzzy quasi-metric from a KM-fuzzy partial metric. Finally, after defining the notion of partial pseudo-metric systems, a one-to-one correspondence between partial pseudo-metric systems and...

A new form of fuzzy α -compactness

Fu Gui Shi (2006)

Mathematica Bohemica

A new form of α -compactness is introduced in L -topological spaces by α -open L -sets and their inequality where L is a complete de Morgan algebra. It doesn’t rely on the structure of the basis lattice L . It can also be characterized by means of α -closed L -sets and their inequality. When L is a completely distributive de Morgan algebra, its many characterizations are presented and the relations between it and the other types of compactness are discussed. Countable α -compactness and the α -Lindelöf property...

A new Lindelöf space with points G δ

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

We prove that ♦ * implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 ℵ 1 which has points G δ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees

Akira Iwasa, Peter J. Nyikos (2006)

Commentationes Mathematicae Universitatis Carolinae

It is independent of the usual (ZFC) axioms of set theory whether every collectionwise Hausdorff tree is either metrizable or has an uncountable chain. We show that even if we add “or has an Aronszajn subtree,” the statement remains ZFC-independent. This is done by constructing a tree as in the title, using the set-theoretic hypothesis ♦ * , which holds in Gödel’s Constructible Universe.

A note on condensations of C p ( X ) onto compacta

Aleksander V. Arhangel'skii, Oleg I. Pavlov (2002)

Commentationes Mathematicae Universitatis Carolinae

A condensation is a one-to-one continuous mapping onto. It is shown that the space C p ( X ) of real-valued continuous functions on X in the topology of pointwise convergence very often cannot be condensed onto a compact Hausdorff space. In particular, this is so for any non-metrizable Eberlein compactum X (Theorem 19). However, there exists a non-metrizable compactum X such that C p ( X ) condenses onto a metrizable compactum (Theorem 10). Several curious open problems are formulated.

A note on discrete sets

Santi Spadaro (2009)

Commentationes Mathematicae Universitatis Carolinae

We give several partial positive answers to a question of Juhász and Szentmiklóssy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences and their closures with the cardinality of a Hausdorff space, improving known results in the literature.

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