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A remark on the tightness of products

Oleg Okunev (1996)

Commentationes Mathematicae Universitatis Carolinae

We observe the existence of a σ -compact, separable topological group G and a countable topological group H such that the tightness of G is countable, but the tightness of G × H is equal to 𝔠 .

A result forO2-convergence to be topological in posets

Qingguo Li, Zhiwei Zou (2016)

Open Mathematics

In this paper, the α waybelow relation, which is determined by O2-convergence, is characterized by the order on a poset, and a sufficient and necessary condition for O2-convergence to be topological is obtained.

A reverse viewpoint on the upper semicontinuity of multivalued maps

Marcio Colombo Fenille (2013)

Mathematica Bohemica

For a multivalued map ϕ : Y ⊸ ( X , τ ) between topological spaces, the upper semifinite topology 𝒜 ( τ ) on the power set 𝒜 ( X ) = { A ⊂ X : A ≠ ∅ } is such that ϕ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map ϕ : Y → ( 𝒜 ( X ) , 𝒜 ( τ ) ) . In this paper, we seek a result like this from a reverse viewpoint, namely, given a set X and a topology Γ on 𝒜 ( X ) , we consider a natural topology ℛ ( Γ ) on X , constructed from Γ satisfying ℛ ( Γ ) = τ if Γ = 𝒜 ( τ ) , and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map ϕ : Y ⊸ ( X , ℛ ( Γ ) ) ...

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number α ≤ 2 , a topological group G such that G γ is countably compact for all cardinals γ < α, but G α is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under M A c o u n t a b l e . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from M A c o u n t a b l e . However, the question has remained...

A sufficient condition for maximal resolvability of topological spaces

Jerzy Bienias, Małgorzata Terepeta (2004)

Commentationes Mathematicae Universitatis Carolinae

We show a new theorem which is a sufficient condition for maximal resolvability of a topological space. We also discuss some relationships between various theorems about maximal resolvability.

A survey on topological games and their applications in analysis.

Jiling Cao, Warren B. Moors (2006)

RACSAM

In this survey article we shall summarise some of the recent progress that has occurred in the study of topological games as well as their applications to abstract analysis. The topics given here do not necessarily represent the most important problems from the area of topological games, but rather, they represent a selection of problems that are of interest to the authors.

A tree π -base for ℝ * without cofinal branches

Fernando Hernández-Hernández (2005)

Commentationes Mathematicae Universitatis Carolinae

We prove an analogue to Dordal’s result in P.L. Dordal, A model in which the base-matrix tree cannot have cofinal branches, J. Symbolic Logic 52 (1980), 651–664. He obtained a model of ZFC in which there is a tree π -base for ℕ * with no ω 2 branches yet of height ω 2 . We establish that this is also possible for ℝ * using a natural modification of Mathias forcing.

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