Extremal phenomena in certain classes of totally bounded groups.

*(English)*Zbl 0703.22002Recall that a space \(X\) is said to be \(\omega\)-bounded [respectively, countably compact] if \(\bar A\) is compact [resp., \(\bar A\setminus A\neq \emptyset\)] for every countable subset \(A\subseteq X\), and that it is called pseudocompact if every continuous function \(X\to\mathbb R\) is bounded. In the following review, the word group always means Hausdorff topological group.

Not surprisingly, the paper deals with cardinal invariants, such as the weight \(\omega(X)\) and the density character \(d(X)\) of \(X\). For an infinite compact group \(K\) it is known that \(| K| =2^{\omega (K)}\), \(d(K)=\log (\omega (K))\). A group is precompact iff it has a compact completion of the same weight with the natural universal property. For a group \(G\) the following implications hold: compact \(\Rightarrow\) \(\omega\)-bounded \(\Rightarrow\) countably compact \(\Rightarrow\) pseudocompact \(\Rightarrow\) precompact.

The monograph is devoted, for each of the possible implications \(P\Rightarrow Q\) among these, to the discussion of the following questions:

(1) Does there exist, for every group topology \(\mathcal T\) on \(G\) satisfying \(P\) a properly finer group topology \(\mathcal T'\supset\mathcal T\) satisfying \(Q\)?

(2) Must a group \(H\) satisfying \(P\) have a proper dense subgroup \(G\) satisfying \(Q\)?

For 15 pairs \((P,Q)\) this gives 30 questions, most of which have negative answers. In particular, since, by a theorem of van der Waerden, the topology of each connected compact simple Lie group is the only precompact one, the 15 answers to question (1) are no. Simple noncompact connected Lie groups have no nonconstant almost periodic functions, hence no precompact topologies. The question (1) remains, nevertheless, meaningful for abelian groups which thus remain at the focus. In particular, for abelian groups, the questions posed almost reduce to the following questions:

(i) Does a pseudocompact group topology of uncountable weight on an abelian group properly enlarge to a pseudocompact one?

(ii) Does every pseudocompact abelian group of uncountable weight have a proper dense pseudocompact subgroup?

The other questions essentially reduce to issues on pseudocompact groups to which most of the monograph is devoted. For a family \(\{G_ i: i\in I\}\) of groups the authors write \(\sum \{\prod_{i\in I}G_ i)\) for the subspace of \(\prod_{i\in I}G_ i\) consisting of all \(x=(x_ i)_{i\in I}\) with \(x_ i=1\) for all but countably finitely many \(i\in I\). For a nonsingleton compact metrizable group \(K\) and an infinite cardinal \(\alpha\) the group \(\sum (K^{\alpha})\) is \(\omega\)-bounded of weight \(\alpha\) and has no proper dense countably compact subgroup. Every (discrete) abelian group has a precompact group topology making it into a group \(G\) without proper dense subgroups. (Take the almost periodic compactification topology in which every subgroup of \(G\) is closed.)

The ample collection of accumulated techniques and constructions does not make for easy reviewing. The reader will notice the authors’ successful effort to outline clearly their objectives at each turn and to evaluate and comment that which has been achieved. Open problems are formulated, partial solutions are explicated in detail. We sample one of the main constructs:

Theorem. Let \(h: G\to H\) denote an algebraic homomorphism between nonsingleton pseudo-compact groups whose graph is a \(G_{\delta}\)-dense subset of \(G\times H\). Then the topology of \(G\) allows a proper pseudocompact refinement of the same weight and \(G\) contains a dense pseudocompact proper subgroup of the same weight. As an example one may take, for any nonsingleton pseudocompact group \(H\) for \(G\) an uncountable power \(H^{\alpha}\) and for \(h: G\to \bar H\) the evaluation of a function \(f: \alpha\to H\) at a point of the Stone-Čech compactification of \(\alpha\) which is represented by a uniform ultrafilter.

The authors call a group extremal if it is pseudocompact and if it has no dense pseudocompact proper subgroup or the topology of \(G\) has no proper refinement to a pseudocompact group topology. A pseudocompact group of countable weight is extremal. Here then is one of the main results of the monograph:

Theorem. A pseudocompact zero-dimensional abelian group of uncountable weight cannot be extremal. In particular, a pseudocompact abelian torsion group of infinite weight cannot be extremal.

Not surprisingly, the paper deals with cardinal invariants, such as the weight \(\omega(X)\) and the density character \(d(X)\) of \(X\). For an infinite compact group \(K\) it is known that \(| K| =2^{\omega (K)}\), \(d(K)=\log (\omega (K))\). A group is precompact iff it has a compact completion of the same weight with the natural universal property. For a group \(G\) the following implications hold: compact \(\Rightarrow\) \(\omega\)-bounded \(\Rightarrow\) countably compact \(\Rightarrow\) pseudocompact \(\Rightarrow\) precompact.

The monograph is devoted, for each of the possible implications \(P\Rightarrow Q\) among these, to the discussion of the following questions:

(1) Does there exist, for every group topology \(\mathcal T\) on \(G\) satisfying \(P\) a properly finer group topology \(\mathcal T'\supset\mathcal T\) satisfying \(Q\)?

(2) Must a group \(H\) satisfying \(P\) have a proper dense subgroup \(G\) satisfying \(Q\)?

For 15 pairs \((P,Q)\) this gives 30 questions, most of which have negative answers. In particular, since, by a theorem of van der Waerden, the topology of each connected compact simple Lie group is the only precompact one, the 15 answers to question (1) are no. Simple noncompact connected Lie groups have no nonconstant almost periodic functions, hence no precompact topologies. The question (1) remains, nevertheless, meaningful for abelian groups which thus remain at the focus. In particular, for abelian groups, the questions posed almost reduce to the following questions:

(i) Does a pseudocompact group topology of uncountable weight on an abelian group properly enlarge to a pseudocompact one?

(ii) Does every pseudocompact abelian group of uncountable weight have a proper dense pseudocompact subgroup?

The other questions essentially reduce to issues on pseudocompact groups to which most of the monograph is devoted. For a family \(\{G_ i: i\in I\}\) of groups the authors write \(\sum \{\prod_{i\in I}G_ i)\) for the subspace of \(\prod_{i\in I}G_ i\) consisting of all \(x=(x_ i)_{i\in I}\) with \(x_ i=1\) for all but countably finitely many \(i\in I\). For a nonsingleton compact metrizable group \(K\) and an infinite cardinal \(\alpha\) the group \(\sum (K^{\alpha})\) is \(\omega\)-bounded of weight \(\alpha\) and has no proper dense countably compact subgroup. Every (discrete) abelian group has a precompact group topology making it into a group \(G\) without proper dense subgroups. (Take the almost periodic compactification topology in which every subgroup of \(G\) is closed.)

The ample collection of accumulated techniques and constructions does not make for easy reviewing. The reader will notice the authors’ successful effort to outline clearly their objectives at each turn and to evaluate and comment that which has been achieved. Open problems are formulated, partial solutions are explicated in detail. We sample one of the main constructs:

Theorem. Let \(h: G\to H\) denote an algebraic homomorphism between nonsingleton pseudo-compact groups whose graph is a \(G_{\delta}\)-dense subset of \(G\times H\). Then the topology of \(G\) allows a proper pseudocompact refinement of the same weight and \(G\) contains a dense pseudocompact proper subgroup of the same weight. As an example one may take, for any nonsingleton pseudocompact group \(H\) for \(G\) an uncountable power \(H^{\alpha}\) and for \(h: G\to \bar H\) the evaluation of a function \(f: \alpha\to H\) at a point of the Stone-Čech compactification of \(\alpha\) which is represented by a uniform ultrafilter.

The authors call a group extremal if it is pseudocompact and if it has no dense pseudocompact proper subgroup or the topology of \(G\) has no proper refinement to a pseudocompact group topology. A pseudocompact group of countable weight is extremal. Here then is one of the main results of the monograph:

Theorem. A pseudocompact zero-dimensional abelian group of uncountable weight cannot be extremal. In particular, a pseudocompact abelian torsion group of infinite weight cannot be extremal.

Reviewer: Karl Heinrich Hofmann (Darmstadt)

##### MSC:

22A05 | Structure of general topological groups |

54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |

22C05 | Compact groups |

22E20 | General properties and structure of other Lie groups |

54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |