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We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in ${[ω]}^{ω}$ , and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.

The Arkhangel'skiĭ–Tall problem under Martin’s Axiom

Gary Gruenhage, Piotr Koszmider (1996)

Fundamenta Mathematicae

We show that MA $_{σ - c e n t e r e d} (ω_{1})$ implies that normal locally compact metacompact spaces are paracompact, and that MA( $ω_{1}$ ) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.

The boolean prime ideal theorem holds iff maximal open filters exist

Y. T. Rhineghost (2002)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

The characterizations of upper approximation operators based on special coverings

Pei Wang, Qingguo Li (2017)

Open Mathematics

In this paper, we discuss the approximation operators [...] apr¯NS ${\bar{a p r}}_{N S}$ and [...] apr¯S ${\bar{a p r}}_{S}$ which are based on NS(U) and S. We not only obtain some properties of NS(U) and S, but also give examples to show some special properties. We also study sufficient and necessary conditions when they become closure operators. In addition, we give general and topological characterizations of the covering for two types of covering-based upper approximation operators being closure operators.

The closures of arithmetic progressions in the common division topology on the set of positive integers

Paulina Szczuka (2014)

Open Mathematics

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.

The common division topology on $ℕ$

José del Carmen Alberto-Domínguez, Gerardo Acosta, Maira Madriz-Mendoza (2022)

Commentationes Mathematicae Universitatis Carolinae

A topological space $X$ is totally Brown if for each $n \in ℕ ∖ {1}$ and every nonempty open subsets $U_{1}, U_{2}, ..., U_{n}$ of $X$ we have ${cl}_{X} (U_{1}) \cap {cl}_{X} (U_{2}) \cap \dots \cap {cl}_{X} (U_{n}) \neq \emptyset$ . Totally Brown spaces are connected. In this paper we consider a topology $τ_{S}$ on the set $ℕ$ of natural numbers. We then present properties of the topological space $(ℕ, τ_{S})$ , some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by...

The convergence space of minimal usco mappings

R. Anguelov, O. F. K. Kalenda (2009)

Czechoslovak Mathematical Journal

A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all...

The counterparts of some cardinal functions in bitopological spaces. II.

Murat Diker (1995)

Mathematica Bohemica

In this paper, bitopological counterparts of the cardinal functions Lindelof number, weak Lindelof number and spread are introduced and studied. Some basic relations between these functions and the functions in [3] are given.

The counterparts of some cardinal functions in bitopological spaces. I.

Murat Diker (1995)

Mathematica Bohemica

This is the first in a series of papers aimed at defining and studying bitopological counterparts of the principal cardinal invariants in topology. It is devoted to study of analogues of the functions weight, density and cellularity.

The decomposition series of a convergence space

Darrell C. Kent, Gary D. Richardson (1973)

Czechoslovak Mathematical Journal

The essential cover and the absolute cover of a schematic space

Wolfgang Rump, Yi Chuan Yang (2009)

Colloquium Mathematicae

A theorem of Gleason states that every compact space admits a projective cover. More generally, in the category of topological spaces with continuous maps, covers exist with respect to the full subcategory of extremally disconnected spaces. Such a cover of a space is called its absolute. We prove that the absolute exists within the category of schematic spaces, i.e. the spaces underlying a scheme. For a schematic space, we use the absolute to generalize Bourbaki's concept of irreducible component,...

The family of $I$ -density type topologies

Grazyna Horbaczewska (2005)

Commentationes Mathematicae Universitatis Carolinae

We investigate a family of topologies introduced similarly as the $I$ -density topology. In particular, we compare these topologies with respect to inclusion and we look for conditions under which these topologies are identical.

The frame of fibrewise closed nuclei

M. Jibladze, P. T. Johnstone (1991)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

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