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On countable families of sets without the Baire property

Mats Aigner, Vitalij A. Chatyrko, Venuste Nyagahakwa (2013)

Colloquium Mathematicae

We suggest a method of constructing decompositions of a topological space X having an open subset homeomorphic to the space (ℝⁿ,τ), where n is an integer ≥ 1 and τ is any admissible extension of the Euclidean topology of ℝⁿ (in particular, X can be a finite-dimensional separable metrizable manifold), into a countable family ℱ of sets (dense in X and zero-dimensional in the case of manifolds) such that the union of each non-empty proper subfamily of ℱ does not have the Baire property in X.

On derivations of quantales

Qimei Xiao, Wenjun Liu (2016)

Open Mathematics

A quantale is a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins. We define the notions of right (left, two) sided derivation and idempotent derivation and investigate the properties of them. It’s well known that quantic nucleus and quantic conucleus play important roles in a quantale. In this paper, the relationships between derivation and quantic nucleus (conucleus) are studied via introducing the concept of pre-derivation.

On Manes' countably compact, countably tight, non-compact spaces

James Dabbs (2011)

Commentationes Mathematicae Universitatis Carolinae

We give a straightforward topological description of a class of spaces that are separable, countably compact, countably tight and Urysohn, but not compact or sequential. We then show that this is the same class of spaces constructed by Manes [Monads in topology, Topology Appl. 157 (2010), 961--989] using a category-theoretical framework.

On minimal strongly KC-spaces

Weihua Sun, Yuming Xu, Ning Li (2009)

Czechoslovak Mathematical Journal

In this article we introduce the notion of strongly KC -spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space ( X , τ ) is maximal countably compact if and only if it is minimal strongly KC , and apply this result to study some properties of minimal strongly KC -spaces, some of which are not possessed by minimal KC -spaces. We also give a positive answer to a question proposed by O. T. Alas and R. G. Wilson, who asked whether every...

On minimal- α -spaces

Giovanni Lo Faro, Giorgio Nordo, Jack R. Porter (2003)

Commentationes Mathematicae Universitatis Carolinae

An α -space is a topological space in which the topology is generated by the family of all α -sets (see [N]). In this paper, minimal- α 𝒫 -spaces (where 𝒫 denotes several separation axioms) are investigated. Some new characterizations of α -spaces are also obtained.

On similarity between topologies

Artur Bartoszewicz, Małgorzata Filipczak, Andrzej Kowalski, Małgorzata Terepeta (2014)

Open Mathematics

Let T 1 and T 2 be topologies defined on the same set X and let us say that (X, T 1) and (X, T 2) are similar if the families of sets which have nonempty interior with respect to T 1 and T 2 coincide. The aim of the paper is to study how similar topologies are related with each other.

On sub-, pseudo- and quasimaximal spaces

J. Schröder (1998)

Commentationes Mathematicae Universitatis Carolinae

The structure of sub-, pseudo- and quasimaximal spaces is investigated. A method of constructing non-trivial quasimaximal spaces is presented.

On the ideal (v 0)

Piotr Kalemba, Szymon Plewik, Anna Wojciechowska (2008)

Open Mathematics

The σ-ideal (v 0) is associated with the Silver forcing, see [5]. Also, it constitutes the family of all completely doughnut null sets, see [9]. We introduce segment topologies to state some resemblances of (v 0) to the family of Ramsey null sets. To describe add(v 0) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v 0) = add(v 0) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v 0) = ω 1 implies that (v 0) has the ideal...

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