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.
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).
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.
Under $\mathfrak{p} = \mathfrak{c}$, we answer Question 24 of \cite{dikranjan&shakhmatov3} for cardinality ${\mathfrak c}$ , by showing that if a non-torsion Abelian group of size continuum admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.
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.
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^{+…
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.
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 $…
We investigate the possibility of the existence of nonsparse strongly summable ultrafilters on certain abelian groups. In particular, we show that every strongly summable ultrafilter on the countably infinite Boolean group is sparse. This answers a question of Hindman, Steprāns and Strauss.
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.
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.
The Wallace problem and countably compact torsion-free Abelian groups in ZFC
Independent researcher（独立研究者）；Department of Mathematics, Institute of Mathematics, Statistics and Computer Science, University of São Paulo（圣保罗大学数学、统计与计算机科学学院）
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.
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 …
Contents: 1. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, I; 2. Frechet-Urysohn fans in free topological groups; 3. Packing index of subsets in Polish groups; 4. Symmetric monochromatic subsets in colorings of the Lobachevsky plane; 5. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups; 6. Distinguishing Number of Countable Homogeneous Relational Structures; 7. Indestructible colourings and rainbow Ramsey theorems; 8. Products of Borel subgroups; 9. Selection theorems and treeabili…
For any countable torsion subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $τ$ such that $H$ is the von Neumann radical of $(G,τ)$. In particular, any unbounded torsion countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. If $G$ is a bounded torsion countably infinite Abelian group, then it admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. In such a case, a MinAP group topology can be chosen to be complete.
CONTENTS: New reals: Can live with them, can live without them; Uniform almost everywhere domination; Heredity of tau-pseudocompactness; Understanding preservation theorems: omega^omega-bounding; Classification problems in continuum theory; Residuality of families of F_sigma sets; First countable, countably compact spaces and the continuum hypothesis; Game approach to universally Kuratowski-Ulam spaces; Small Valdivia compact spaces; Canonical forms of Borel functions on the Milliken space; Complete analytic equivalence relations; The Number of Near-Coherence Classes of Ultrafilters is Either …
Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We say that the generic countably infinite group is of property $P$ if $P$ is comeager in $G$. We prove that every group property with the Baire property is either meager or comeager. We show that there is a comeager elementary equivalence class in $G$ but every isomorphism class is meager. We prove that the generic group is algebraically closed, simple, not fin…
The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent sequences having the following "selective" compactness property: For each free ultrafilter p on N and every sequence {U_n:n in N} of non-empty open subsets of G one can choose a point x_n in U_n for all n in such a way that the resulting sequence {x_n:n in N} has a p-limit in G, that is, {n in N: x_n in V} belongs to p for every neighbourhood V of x in G. In par…
