On spaces with the ideal convergence property

Jakub Jasinski; Ireneusz Recław

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Displaying similar documents to “On spaces with the ideal convergence property”

Fixed-place ideals in commutative rings

Ali Rezaei Aliabad, Mehdi Badie (2013)

Commentationes Mathematicae Universitatis Carolinae

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Let $I$ be a semi-prime ideal. Then $P_{\circ} \in Min (I)$ is called irredundant with respect to $I$ if $I \neq ⋂_{P_{\circ} \neq P \in Min (I)} P$ . If $I$ is the intersection of all irredundant ideals with respect to $I$ , it is called a fixed-place ideal. If there are no irredundant ideals with respect to $I$ , it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point $p \in β X$ is a fixed-place point if $O^{p} (X)$ is a fixed-place ideal. In...

Normal restrictions of the noncofinal ideal on $P_{κ} (λ)$

Pierre Matet (2013)

Fundamenta Mathematicae

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We discuss the problem of whether there exists a restriction of the noncofinal ideal on $P_{κ} (λ)$ that is normal.

A characterization of the meager ideal

Piotr Zakrzewski (2015)

Commentationes Mathematicae Universitatis Carolinae

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We give a classical proof of the theorem stating that the $σ$ -ideal of meager sets is the unique $σ$ -ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.

The ideal (a) is not $G_{δ}$ generated

Marta Frankowska, Andrzej Nowik (2011)

Colloquium Mathematicae

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We prove that the ideal (a) defined by the density topology is not $G_{δ}$ generated. This answers a question of Z. Grande and E. Strońska.

Ideal convergence and divergence of nets in $(ℓ)$ -groups

Antonio Boccuto, Xenofon Dimitriou, Nikolaos Papanastassiou (2012)

Czechoslovak Mathematical Journal

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In this paper we introduce the $ℐ$ - and $ℐ^{*}$ -convergence and divergence of nets in $(ℓ)$ -groups. We prove some theorems relating different types of convergence/divergence for nets in $(ℓ)$ -group setting, in relation with ideals. We consider both order and $(D)$ -convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that $ℐ^{*}$ -convergence/divergence implies $ℐ$ -convergence/divergence for every ideal,...

On nonregular ideals and $z^{\circ}$ -ideals in $C (X)$

F. Azarpanah, M. Karavan (2005)

Czechoslovak Mathematical Journal

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The spaces $X$ in which every prime $z^{\circ}$ -ideal of $C (X)$ is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces $X$ , such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime $z^{\circ}$ -ideal in $C (X)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C (X)$ a $z^{\circ}$ -ideal? When is every nonregular (prime) $z$ -ideal in $C (X)$ a...

Ideal limits of sequences of continuous functions

Miklós Laczkovich, Ireneusz Recław (2009)

Fundamenta Mathematicae

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We prove that for every Borel ideal, the ideal limits of sequences of continuous functions on a Polish space are of Baire class one if and only if the ideal does not contain a copy of Fin × Fin. In particular, this is true for $F_{σ δ}$ ideals. In the proof we use Borel determinacy for a game introduced by C. Laflamme.

Intersections of essential minimal prime ideals

A. Taherifar (2014)

Commentationes Mathematicae Universitatis Carolinae

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Let $𝒵 (ℛ)$ be the set of zero divisor elements of a commutative ring $R$ with identity and $ℳ$ be the space of minimal prime ideals of $R$ with Zariski topology. An ideal $I$ of $R$ is called strongly dense ideal or briefly $s d$ -ideal if $I \subseteq 𝒵 (ℛ)$ and $I$ is contained in no minimal prime ideal. We denote by $R_{K} (ℳ)$ , the set of all $a \in R$ for which $\bar{D (a)} = \bar{ℳ ∖ V (a)}$ is compact. We show that $R$ has property $(A)$ and $ℳ$ is compact if and only if $R$ has no $s d$ -ideal. It is proved that $R_{K} (ℳ)$ is an essential ideal (resp., $s d$ -ideal) if and only if $ℳ$ is an almost...

When does the Katětov order imply that one ideal extends the other?

Paweł Barbarski, Rafał Filipów, Nikodem Mrożek, Piotr Szuca (2013)

Colloquium Mathematicae

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We consider the Katětov order between ideals of subsets of natural numbers (" $\leq_{K}$ ") and its stronger variant-containing an isomorphic ideal ("⊑ "). In particular, we are interested in ideals for which $\leq_{K} \Rightarrow ⊑$ for every ideal . We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order "⊑ " (and vice versa).