On the Suslin number of subgroups of products of countable groups. (English) Zbl 0854.20064

The Suslin number \(c(X)\) of a topological space \(X\) is defined as follows: if \(\kappa\) is a cardinal, then \(c(X)\leq\kappa\) iff any family of pairwise disjoint non-empty open sets in \(X\) has cardinality \(\leq\kappa\). A. V. Arkhangel’skij asked if there exists a subgroup \(G\) of \(\mathbb{Z}^{\mathfrak c}\) such that \(c(G)={\mathfrak c}=2^\omega\). The author gives an affirmative answer, the proof being based on the following Lemma: Let \(I\) be a set of cardinality \(\mathfrak c\). There exists an \(I\times I\)-matrix \((a_{ij})\) with integer coefficients such that for any distinct \(i,j\in I\) there exists a prime \(p\) such that \(a_{ij}\not\equiv a_{ji}\pmod p\) and \(a_{ih}\equiv a_{jh}\pmod p\) for every \(h\in I-\{i,j\}\).
Remark: The author has also other similar results from 1982.


20K25 Direct sums, direct products, etc. for abelian groups
20K27 Subgroups of abelian groups
20K45 Topological methods for abelian groups
22A05 Structure of general topological groups


Zbl 0527.22001
Full Text: EuDML