×

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.

MSC:

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

Citations:

Zbl 0527.22001
PDFBibTeX XMLCite
Full Text: EuDML