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Artur Hideyuki Tomita

Publications and source records attributed to Artur Hideyuki Tomita.

12 recordsLinked to original sources
arXiv2026

Countable compactness in powers of topological groups and Ramsey theoretic variations of compactness

Countable compactness need not be preserved by finite products. Motivated by compactness notions for maps indexed by finite subsets of $ω$, we introduce cascade countable compactness for arbitrary barriers. For $\mathcal B=[ω]^2$, this is the previously studied notion of being doubly countably compact. We show that, in ZFC, if $G$ is a Hausdorff topological group and $1\leq k<ω$, then $k$-cascade countable compactness of $G$ implies that $G^k$ is countably compact. Cascade countable compactness for the Schreier barrier implies that $G^ω$ is countably compact. In contrast, we construct a Tychonoff space that is $n$-cascade countably compact for every $n<ω$ but has a non-countably compact square, and a Hausdorff Boolean group $H$ that is $\mathcal B$-countably compact for every barrier $\mathcal B$ but whose square is not countably compact. We also construct a Hausdorff Boolean group without nontrivial convergent sequences that is $\mathcal B$-cascade countably compact for every barrier $\mathcal B$. Finally, we obtain a subspace $X\subseteqβω$ such that $X^κ$ is $n$-cascade countably compact for every $κ<\mathfrak h$ and every $n<ω$, whereas $\exp X$ is not pseudocompact.

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arXiv2024

Remarks on SHD spaces and more divergence properties

The class of SHD spaces was recently introduced in [12]. The first part of this paper focuses on answering most of the questions presented in that article. For instance, we exhibit an example of a non-SHD Tychonoff space $X$ such that $\mathscr{F}[X]$, the Pixley-Roy hyperspace of $X$, $βX$, the Stone-Čech compactification of $X$, and $C_p(X)$, the ring of continuous functions over $X$ equipped with the topology of pointwise convergence, are SHD. In the second part of this work we will present some variations of the SHD notion, namely, the WSHD property and the OHD property. Furthermore, we will pay special attention to the relationships between $X$ and $\mathscr{F}[X]$ regarding these new concepts.

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arXiv2024

A Wallace semigroup whose every finite power is countably compact

We show that, assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, there exists a Wallace semigroup whose infinite countable power is the least power which fails to be countably compact. This answers positively Question 9.4 of \cite{Tomita15}.

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arXiv2022

On powers of countably pracompact groups

In 1990, Comfort asked: is there, for every cardinal number $α\leq 2^{\mathfrak{c}}$, a topological group $G$ such that $G^γ$ is countably compact for all cardinals $γ<α$, but $G^α$ is not countably compact? A similar question can also be asked for countably pracompact groups: for which cardinals $α$ is there a topological group $G$ such that $G^γ$ is countably pracompact for all cardinals $γ< α$, but $G^α$ is not countably pracompact? In this paper we construct such group in the case $α= ω$, assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, and in the case $α= κ^{+}$, with $ω\leq κ\leq 2^{\mathfrak{c}}$, assuming the existence of $2^{\mathfrak{c}}$ incomparable selective ultrafilters. In particular, under the second assumption, there exists a topological group $G$ so that $G^{2^\mathfrak{c}}$ is countably pracompact, but $G^{(2^{\mathfrak{c}})^{+}}$ is not countably pracompact, unlike the countably compact case.

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arXiv2022

$\mathcal{U}$-compact group topologies without convergent sequences on countably cofinal torsion-free Abelian groups

We obtain a forcing construction that shows that it is consistent that the torsion-free Abelian group $\mathbb{Q}^{(λ)}$ admits a Hausdorff group topology which is also $\mathcal{U}$-compact and contains no non-trivial convergent sequences, where $λ$ is a cardinal whose cofinality is $ω$ and $\mathcal{U}$ is a selective ultrafilter. This answers a question posed in arXiv:1904.05928.

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arXiv2021

Some pseudocompact-like properties in certain topological groups

We construct in ZFC a countably compact group without non-trivial convergent sequences of size $2^{\mathfrak{c}}$, answering a question of Bellini, Rodrigues and Tomita. We also construct in ZFC a selectively pseudocompact group which is not countably pracompact, showing that these two properties are not equivalent in the class of topological groups. Using the same technique, we construct a group which has all powers selectively pseudocompact but is not countably pracompact, assuming the existence of a selective ultrafilter.

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arXiv2020

Forcing a classification of non-torsion Abelian groups of size at most $2^\mathfrak c$ with non-trivial convergent sequences

We force a classification of all the Abelian groups of cardinality at most $2^\mathfrak c$ that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov for cardinality at most $2^{\mathfrak c}$, by showing that if a non-torsion Abelian group of size at most $2^\mathfrak c$ admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.

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arXiv2020

Maximal almost disjoint families and pseudocompactness of hyperspaces

We show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if $\mathsf{MA}_\mathfrak c (\mathcal P(ω)/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost disjoint family whose hyperspace is pseudocompact and prove that consistently such families do not exist \emph{genericaly}, by constructing a consistent example of a maximal almost disjoint family $\mathcal A$ of size less than $\mathfrak c$ whose hyperspace is not pseudocompact.

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arXiv2019

A van Douwen-like ZFC theorem for small powers of countably compact groups without non-trivial convergent sequences

We show that if $κ\leq ω$ and there exists a group topology without non-trivial convergent sequences on an Abelian group $H$ such that $H^n$ is countably compact for each $n<κ$ then there exists a topological group $G$ such that $G^n$ is countably compact for each $n <κ$ and $G^κ$ is not countably compact. If in addition $H$ is torsion, then the result above holds for $κ=ω_1$. Combining with other results in the literature, we show that: $a)$ Assuming ${\mathfrak c}$ incomparable selective ultrafilters, for each $n \in ω$, there exists a group topology on the free Abelian group $G$ such that $G^n$ is countably compact and $G^{n+1}$ is not countably compact. (It was already know for $ω$). $b)$ If $κ\in ω\cup \{ω\} \cup \{ω_1\}$, there exists in ZFC a topological group $G$ such that $G^γ$ is countably compact for each cardinal $γ<κ$ and $G^κ$ is not countably compact.

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arXiv2019

On $p$-compact group topologies on direct sums of ${\mathbb Q}$

We prove that if $p$ is a selective ultrafilter then ${\mathbb Q}^{(κ)}$ has a $p$-compact group topology without non-trivial convergent sequences, for each infinite cardinal $κ=κ^ω$. In particular, this gives the first arbitrarily large examples of countably compact groups without non-trivial convergent sequences that are torsion-free.

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arXiv2018

Small MAD families whose Isbell-Mrówka spaces are pseudocompact

Given a countable transitive model $M$ for ZFC+CH, we prove that one can produce a maximal almost disjoint family in $M$ whose Vietoris Hyperspace of its Isbell-Mrówka space is pseudocompact on every Cohen extension of $M$. We also show that a classical example of $ω_1$-sized maximal almost disjoint family obtained by a forcing iteration of length $ω_1$ in a model of non CH is such that the Vietoris Hyperspace of its Isbell-Mrówka space is pseudocompact.

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