Countably compact groups without non-trivial convergent sequences

The celebrated Comfort-Ross theorem [11, 7] states that any product of pseudo-compact topological groups is pseudo-compact, in stark contrast with the examples due to Novák [28] and Terasaka [34] who constructed pairs of countably compact spaces whose product is not even pseudo-compact. This motivated Comfort [9] (repeated in [8]) to ask:

The first consistent positive answer was given by van Douwen [45] under MA, followed by Hart-van Mill [21] under MA_ctble. In his paper van Douwen showed that every Boolean countably compact group without non-trivial convergent sequences contains two countably compact subgroups whose product is not countably compact, and asked:

In fact, the first example of such a group was constructed by Hajnal and Juhász [20] a few years before van Douwen’s [45] assuming CH. Recall, that every compact topological group contains a non-trivial convergent sequence, as an easy consequence of the classical and highly non-trivial theorem of Ivanovskiĭ-Vilenkin-Kuz’minov (see [25]) that every compact topological group is dyadic, i.e., a continuous image of $2^{\kappa}$ for some cardinal number $\kappa$.

Both questions have been studied extensively in recent decades, providing a large variety of sufficient conditions for the existence of examples to these questions, much work being done by Tomita and collaborators [17, 18, 23, 30, 33, 40, 41, 42, 37, 43, 44, 38], but also others [10, 13, 14, 27, 35]. The questions are considered central in the theory of topological groups [1, 2, 7, 8, 15, 32, 36].

Here we settle both problems by constructing in ZFC a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences.

The paper is organized as follows: In Section 2 we fix notation and review basic facts concerning ultrapowers, Fubini products of ultrafilters and Bohr topology. In Section 3 we study van Douwen’s problem in the realm of $p$-compact groups. We show how iterated ultrapowers can be used to give interesting partial solutions to the problem. In particular, we show that an iterated ultrapower of the countable Boolean group endowed with the Bohr topology via a selective ultrafilter $p$ produces a $p$-compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences. This on the one hand raises interesting questions about ultrafilters, and on the other hand serves as a warm up for Section 4, where the main result of the paper is proved by constructing a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences using not a single ultrafilter, but rather a carefully constructed $\mathfrak{c}$-sized family of ultrafilters.

Recall that an infinite topological space $X$ is countably compact if every infinite subset of $X$ has an accumulation point. Given $p$ a nonprincipal ultrafilter on $\omega$ (for short, $p\in\omega^{*}$), a point $x\in X$ and a sequence $\{x_{n}:n\in\omega\}\subseteq X$ we say (following [5]) that $x=p$-$\lim_{n\in\omega}x_{n}$ if for every open $U\subseteq X$ containing $x$ the set $\{n\in\omega\colon x_{n}\in U\}\in p$. It follows that a space $X$ is countably compact if and only if every sequence $\{x_{n}:n\in\omega\}\subseteq X$ has a $p$-limit in $X$ for some ultrafilter $p\in\omega^{*}$. Given an ultrafilter $p\in\omega^{*}$, a space $X$ is $p$-compact if for every sequence $\{x_{n}:n\in\omega\}\subseteq X$ there is an $x\in X$ such that $x=p$-$\lim_{n\in\omega}x_{n}$.

For introducing the following definition, we fix a bijection $\varphi:\omega\to\omega\times\omega$, and for a limit ordinal $\alpha<\omega_{1}$, we pick an increasing sequence $\{\alpha_{n}:n\in\omega\}$ of smaller ordinals with supremum $\alpha$. Given an ultrafilter $p\in\omega^{*}$, the iterated Fubini powers or Frolík sums [16] of $p$ are defined recursively as follows:

The choice of the ultrafilter $p^{\alpha}$ depends on (the arbitrary) choice of $\varphi$ and the choice of the sequence $\{\alpha_{n}:n\in\omega\}$, however, the type of $p^{\alpha}$ does not (see e.g., [16, 19]).

For our purposes we give an alternative definition of the iterated Fubini powers of $p$: given $\alpha<\omega_{1}$ we fix a well-founded tree $T_{\alpha}\subset\omega^{<\omega}$ such that

1. (i)

   $\rho_{T_{\alpha}}(\varnothing)=\alpha$, where $\rho_{T_{\alpha}}$ denotes the rank function on $\langle T_{\alpha},\subseteq\rangle$;

2. (ii)

   For every $t\in T_{\alpha}$, if $\rho_{T_{\alpha}}(t)>0$ then $t^{\frown}n\in T_{\alpha}$ for all $n\in\omega$.

For $\beta\leqslant\alpha$, let $\Omega_{\beta}(T_{\alpha})=\{t\in T_{\alpha}\colon\rho_{T_{\alpha}}(t)=\beta\}$ and $T_{\alpha}^{+}=\{t\in T_{\alpha}\colon\rho_{T_{\alpha}}(t)>0\}$.

If $p\in\omega^{*}$, then $\mathbb{L}_{p}(T_{\alpha})$ will be used to denote the collection of all trees $T\subseteq T_{\alpha}$ such that for every $t\in T\cap T_{\alpha}^{+}$ the set $\text{succ}_{T}(t)=\{n\in\omega\colon t^{\frown}n\in T\}$ belongs to $p$. Notice that each $T\in\mathbb{L}_{p}(T_{\alpha})$ is also a well-founded tree with $\rho_{T}(\varnothing)=\alpha$. Moreover, the family $\{\Omega_{0}(T)\colon T\in\mathbb{L}_{p}(T_{\alpha})\}$ forms a base of an ultrafilter on $\Omega_{0}(T_{\alpha})$ which has the same type of $p^{\alpha}$. If $T\in\mathbb{L}_{p}(T_{\alpha})$ and $U\in p$, $T\upharpoonright{U}$ denotes the tree in $\mathbb{L}_{p}(T_{\alpha})$ for which $\text{succ}_{T\upharpoonright{U}}(t)=\text{succ}_{T}(t)\cap U$ for all $t\in(T\upharpoonright{U})^{+}$.

Next we recall the ultrapower construction from model theory and algebra. Given a group $\mathbb{G}$ and an ultrafilter $p\in\omega^{*}$, denote by

The Theorem of Łós [26] states that for any formula $\phi$ with parameters $[f_{0}],[f_{1}],\dots$ $[f_{n}]$, $\mathsf{ult}_{p}(\mathbb{G})\models\phi([f_{0}],[f_{1}],\dots[f_{n}])$ if and only if $\{k:\mathbb{G}\models\phi(f_{0}(k),f_{1}(k),$ $\dots$ $f_{n}(k))\}\in p$. In particular, $\mathsf{ult}_{p}(\mathbb{G})$ is a group with the same first order properties as $\mathbb{G}$.

There is a natural embedding of $\mathbb{G}$ into $\mathsf{ult}_{p}(\mathbb{G})$ sending each $g\in\mathbb{G}$ to the equivalence class of the constant function with value $g$. We shall therefore consider $\mathbb{G}$ as a subgroup of $\mathsf{ult}_{p}(\mathbb{G})$. Also, without loss of generality, we can assume that $\text{dom}(f)\in p$ for every $[f]\in\mathsf{ult}_{p}(\mathbb{G})$.

Recall that the Bohr topology on a group $\mathbb{G}$ is the weakest group topology making every homomorphism $\Phi\in\text{Hom}(\mathbb{G},\mathbb{T})$ continuous, where the circle group $\mathbb{T}$ carries the usual compact topology. We let $(\mathbb{G},\tau_{\,\text{Bohr}})$ denote $\mathbb{G}$ equipped with the Bohr topology.

Finally, our set-theoretic notation is mostly standard and follows [24]. In particular, recall that an ultrafilter $p\in\omega^{*}$ is a $P$-point if every function on $\omega$ is finite-to-one or constant when restricted to some set in the ultrafilter and, an ultrafilter $p\in\omega^{*}$ is a $Q$-point if every finite-to-one function on $\omega$ becomes one-to-one when restricted to a suitable set in the ultrafilter. The ultrafilters $p\in\omega^{*}$ which are P-point and Q-point are called selective ultrafilters. For more background on set-theoretic aspects of ultrafilters see [6].

In this section we shall give a canonical construction of a $p$-compact group for every ultrafilter $p\in\omega^{*}$. This will be done by studying the iterated ultrapower construction.

Fix a group $\mathbb{G}$ and put $\mathsf{ult}_{p}^{0}(\mathbb{G})=\mathbb{G}$. Given an ordinal $\alpha$ with $\alpha>0$, let

where $\varinjlim_{\beta<\alpha}\mathsf{ult}_{p}^{\beta}(\mathbb{G})$ denotes the direct limit of the direct system $\langle\mathsf{ult}_{p}^{\beta}(\mathbb{G}),\varphi_{\delta\beta}\colon\delta\leqslant\beta<\alpha\rangle$ with the following properties:

1. (1)

   $\varphi_{\delta\delta}$ is the identity function on $\mathsf{ult}_{p}^{\delta}(\mathbb{G})$, and

2. (2)

   $\varphi_{\delta\beta}\colon\mathsf{ult}_{p}^{\delta}(\mathbb{G})\to\mathsf{ult}_{p}^{\beta}(\mathbb{G})$ is the canonical embedding of $\mathsf{ult}_{p}^{\delta}(\mathbb{G})$ into $\mathsf{ult}_{p}^{\beta}(\mathbb{G})$, defined recursively by $\varphi_{\delta,\alpha+1}([f])=$ the constant function with value $[f]$, and $\varphi_{\delta,\alpha}([f])=$ the direct limit of $\varphi_{\delta,\beta}([f]),\ \beta<\alpha$ for a limit ordinal $\alpha$.

In what follows, we will abbreviate $\mathsf{ult}_{p}^{\alpha^{-}}(\mathbb{G})$ for $\varinjlim_{\beta<\alpha}\mathsf{ult}_{p}^{\beta}(\mathbb{G})$. Moreover, we will treat $\mathsf{ult}_{p}^{\alpha^{-}}(\mathbb{G})$ as $\bigcup_{\beta<\alpha}\mathsf{ult}_{p}^{\beta}(\mathbb{G})$ and, in such case, we put $\text{ht}(a)=\min\{\beta<\alpha\colon a\in\mathsf{ult}_{p}^{\beta}(\mathbb{G})\}$ for every $a\in\mathsf{ult}_{p}^{\alpha^{-}}(\mathbb{G})$. This is, of course, formally wrong, but is facilitated by our identification of $\mathbb{G}$ with a subgroup of $\mathsf{ult}_{p}(\mathbb{G})$. In this way we can avoid talking about direct limit constructions.

We now consider $(\mathbb{G},\tau_{\,\text{Bohr}})$. Having fixed an ultrafilter $p\in\omega^{*}$, this topology naturally lifts to a topology on $\mathsf{ult}_{p}(\mathbb{G})$ as follows: Every $\Phi\in\text{Hom}(\mathbb{G},\mathbb{T})$ naturally extends to a homomorphism $\overline{\Phi}\in\text{Hom}(\mathsf{ult}_{p}(\mathbb{G}),\mathbb{T})$ by letting

By Łós’s theorem, $\overline{\Phi}$ is indeed a homomorphism from $\mathsf{ult}_{p}(\mathbb{G})$ to $\mathbb{T}$ and hence the weakest topology making every $\overline{\Phi}$ continuous, where $\Phi\in\text{Hom}(\mathbb{G},\mathbb{T})$, is a group topology on $\mathsf{ult}_{p}(\mathbb{G})$. This topology will be denoted by $\tau_{\,\overline{\text{Bohr}}}$.

The following is a trivial, yet fundamental fact:

The group that will be relevant for us is the group $\mathsf{ult}_{p}^{\omega_{1}}(\mathbb{G})$, endowed with the topology $\tau_{\,\overline{\text{Bohr}}}$ induced by the homomorphisms in $\text{Hom}(\mathbb{G},\mathbb{T})$ extended recursively all the way to $\mathsf{ult}_{p}^{\omega_{1}}(\mathbb{G})$ by the same formula (3.1).

The (iterated) ultrapower with this topology is usually not Hausdorff (see [12, 3]), so we identify the inseparable functions and denote by $(\mathsf{Ult}_{p}^{\omega_{1}}(\mathbb{G}),\tau_{\,\overline{\text{Bohr}}})$ this quotient. More explicitly,

where $K=\bigcap_{\Phi\in\text{Hom}(\mathbb{G},\mathbb{T})}\text{Ker}(\overline{\Phi})$. The natural projection will be denoted by

The main reason for considering the iterated Fubini powers here is the following simple and crucial fact:

The plan for our construction is as follows: fix an ultrafilter $p\in\omega^{*}$, find a suitable topological group $\mathbb{G}$ without convergent sequences and consider $(\mathsf{Ult}_{p}^{\omega_{1}}(\mathbb{G}),\tau_{\,\overline{\text{Bohr}}})$. The remaining issue is: Does $(\mathsf{Ult}_{p}^{\omega_{1}}(\mathbb{G}),\tau_{\,\overline{\text{Bohr}}})$ have non-trivial convergent sequences?

While our approach is applicable to an arbitrary group $\mathbb{G}$, in the remainder of this paper we will be dealing exclusively with Boolean groups, i.e., groups where each element is its own inverse.¹¹1The general case will be dealt with in a separate paper. These groups are, in every infinite cardinality $\kappa$, isomorphic to the group $[\kappa]^{<\omega}$ with the symmetric difference $\triangle$ as the group operation and $\varnothing$ as the neutral element. Every Boolean group is a vector space over the trivial $2$-element field which we identify with $2=\{0,1\}$. Hence, we can talk, e.g., about linearly independent subsets of a Boolean group. Also, since every homomorphism from a Boolean group into the torus $\mathbb{T}$ takes at most two values (in the unique subgroup of $\mathbb{T}$ of size $2$) we may and will identify $\text{Hom}([\omega]^{<\omega},\mathbb{T})$ with $\text{Hom}([\omega]^{<\omega},2)$ to highlight the fact that there are only two possible values. Hence also $\text{Hom}([\omega]^{<\omega},2)$ is a Boolean group and a vector space over the same field.

The following theorem is the main result of this section.

In order to prove this theorem, we apply the first step of our plan.

We say that a sequence $\langle[f_{n}]\colon n\in\omega\rangle\subset\mathsf{ult}_{p}([\omega]^{<\omega})$ is $p$-separated if for every $n\neq m\in\omega$ there is a $\Phi\in\text{Hom}([\omega]^{<\omega},2)$ such that $\overline{\Phi}([f_{n}])\neq\overline{\Phi}([f_{m}])$. In other words, a sequence $\langle[f_{n}]\colon n\in\omega\rangle\subset\mathsf{ult}_{p}([\omega]^{<\omega})$ is $p$-separated if and only if its elements represent distinct elements of

where $K=\bigcap_{\Phi\in\text{Hom}([\omega]^{<\omega},2)}\text{Ker}(\overline{\Phi})$ and $\pi:\mathsf{ult}_{p}([\omega]^{<\omega})\to\mathsf{Ult}_{p}([\omega]^{<\omega})$ is the corresponding projection.

We next show that, in general, the plan does not work for all $p\in\omega^{*}$.

As $(\mathsf{Ult}_{p}([\omega]^{<\omega}),\tau_{\,\overline{\text{Bohr}}})$ is a topological subgroup of $(\mathsf{Ult}_{p}^{\omega_{1}}([\omega]^{<\omega}),\tau_{\,\overline{\text{Bohr}}})$ there are ultrafilters (even P-points assuming CH) such that $\mathsf{Ult}_{p}^{\omega_{1}}([\omega]^{<\omega})$ has a non-trivial convergent sequence.

Selective ultrafilters and Q-points, have immediate combinatorial reformulations relevant in our context. Given a non-empty set $I$ and $\mathbb{G}$ a Boolean group, we shall call a set $\{f_{i}\colon i\in I\}$ of functions $f_{i}\colon\omega\to\mathbb{G}$ $p$-independent if

for every non-empty finite set $E\subset I$ and every $a\in\mathbb{G}$. Note that, in particular, a function $f\colon\omega\to\mathbb{G}$ is not constant on an element of $p$ if and only if $\{f\}$ is $p$-independent. Now, we will say that a function $f\colon I\to\mathbb{G}$ is linearly independent if $f$ is one-to-one and $\{f(i)\colon i\in I\}$ is a linearly independent set and, a function $f\colon I\to\mathsf{ult}_{p}(\mathbb{G})$ is $p$-independent if $f$ is one-to-one and $\{f_{i}\colon i\in I\}$ is a $p$-independent set, where $f(i)=[f_{i}]$ for $i\in I$.