arXiv subjects

V. O. Rodrigues

Publications and source records attributed to V. O. Rodrigues.

3 recordsLinked to original sources

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↗

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↗