arXiv subjects

A. H. Tomita

Publications and source records attributed to A. H. Tomita.

7 recordsLinked to original sources

arXiv2022

Comfort's question on powers in $\mathbb Q ^{(2^\mathfrak c)}$ and a Wallace semigroup whose cube is countably compact

We prove that the existence of $\mathfrak{c}$ incomparable selective ultrafilters implies the existence of a Wallace semigroup whose cube is countably compact. In addition, assuming the existence of $2^{\mathfrak c}$ incomparable selective ultrafilters and $2^{< 2^{\mathfrak{c}}} = 2^{\mathfrak{c}}$, we obtain torsion-free topological groups with respect to Comfort's question on the countable compactness of (infinite) powers of a topological group.

math.GN↗

arXiv2021

Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

Assuming the existence of $\mathfrak c$ incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality $\mathfrak c$ that admit a countably compact group topology. We show that for each $κ\in [\mathfrak c, 2^\mathfrak c]$ each of these groups has a countably compact group topology of weight $κ$ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of $2^\mathfrak c$ selective ultrafilters, there are at least $2^\mathfrak c$ non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most $2^\mathfrak c$ is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality $\mathfrak c$ that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.

math.GN↗

arXiv2021

Countably compact group topologies on arbitrarily large free Abelian groups

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $κ$ such that $κ^ω=κ$, there exists a group topology on the free Abelian group of cardinality $κ$ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.

math.LO↗

arXiv2019

Countably compact group topologies on the free Abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter

We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).

math.GN↗

arXiv2017

Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $βω$

We study the relations between a generalization of pseudocompactness, named $(κ, M)$-pseudocompactness, the countably compactness of subspaces of $βω$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $\mathfrak c$-many selective ultrafilters, that there exists a subspace of $βω$ that is $(κ, ω^*)$-pseudocompact for all $κ<\mathfrak c$, but $\text{CL}(X)$ isn't pseudocompact. We prove in ZFC that if $ω\subseteq X\subseteq βω$ is such that $X$ is $(\mathfrak c, ω^*)$-pseudocompact, then $\text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $\mathfrak c$ for some small cardinals. We provide an example of a subspace of $βω$ for which all powers below $\mathfrak h$ are countably compact whose hyperspace is not pseudocompact, we show that if $ω\subseteq X$, the pseudocompactness of $\text{CL}(X)$ implies that $X$ is $(κ, ω^*)$-pseudocompact for all $κ<\mathfrak h$, and provide an example of such an $X$ that is not $(\mathfrak b, ω^*)$-pseudocompact.

math.GN↗

arXiv2017

Finite powers of selectively pseudocompact groups

A space $X$ is called {\it selectively pseudocompact} if for each sequence $(U_{n})_{n\in \mathbb{N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{n})_{n\in \mathbb{N}}$ of points in $X$ such that $cl_X(\{x_n : n < ω\}) \setminus \big(\bigcup_{n < ω}U_n \big) \neq \emptyset$ and $x_{n}\in U_{n}$, for each $n < ω$. Countably compact space spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of $CH$, that for every positive integer $k > 2$ there exists a topological group whose $k$-th power is countably compact but its $(k+1)$-st power is not selectively pseudocompact. This provides a positive answer to a question posed in \cite{gt} in any model of $ZFC+CH$.

math.GN↗

arXiv2016

$σ$-Ideals and outer measures on the real line

A {\it weak selection} on $\mathbb{R}$ is a function $f: [\mathbb{R}]^2 \to \mathbb{R}$ such that $f(\{x,y\}) \in \{x,y\}$ for each $\{x,y\} \in [\mathbb{R}]^2$. In this article, we continue with the study (which was initiated in \cite{ag}) of the outer measures $λ_f$ on the real line $\mathbb{R}$ defined by weak selections $f$. One of the main results is to show that $CH$ is equivalent to the existence of a weak selection $f$ for which: \[ \mathcal λ_f(A)= \begin{cases} 0 & \text{if $|A| \leq ω$,}\\ \infty & \text{otherwise.} \end{cases} \] Some conditions are given for a $σ$-ideal of $\mathbb{R}$ in order to be exactly the family $\mathcal{N}_f$ of $λ_f$-null subsets for some weak selection $f$. It is shown that there are $2^\mathfrak{c}$ pairwise distinct ideals on $\mathbb{R}$ of the form $\mathcal{N}_f$, where $f$ is a weak selection. Also we prove that Martin Axiom implies the existence of a weak selection $f$ such that $\mathcal{N}_f$ is exactly the $σ$-ideal of meager subsets of $\mathbb{R}$. Finally, we shall study pairs of weak selections which are "almost equal" but they have different families of $λ_f$-measurable sets.

math.GN↗