Equilateral sets in Banach spaces of the form \(C(K)\) (Q2804301)

A subset \(S\) of a metric space \((M,d)\) is said to be \(\lambda\)-equilateral (\(\lambda>0\)) if \(d(x,y)=\lambda\) whenever \(x\) and \(y\) are different elements of \(S\). \(S\) is said to be equilateral if \(S\) is \(\lambda\)-equilateral for some positive \(\lambda\). An equilateral set \(S\subset M\) is said to be maximal if there is no equilateral set \(B\subset M\) with \(S\subsetneq B\).NEWLINENEWLINEThe authors prove that, given a compact Hausdorff space \(K\) and an infinite cardinal \(\alpha\), the unit ball \(B_{C(K)}\) contains a \((1+\varepsilon)\)-separated set (for some \(\varepsilon>0\)) of size \(\alpha\) if and only if the unit sphere \(S_{C(K)}\) contains a 2-equilateral set of size \(\alpha\), if and only if \(B_{C(K)}\) contains a \(\lambda\)-equilateral set, with \(\lambda>1\), of size \(\alpha\) (Theorem 2.6). Moreover, many different sufficient conditions on a compact Hausdorff space \(K\) are listed (Theorem 2.9) for \(B_{C(K)}\) to contain an uncountable 2-equilateral set. The key for proving this theorem is a combinatorial tool involving linked families of closed pairs, introduced in the first part of the paper.NEWLINENEWLINESome questions are raised in this context: the authors provide references about the updated solutions to two of them and leave one of them open (Remark 2.10). In the last section, maximal equilateral sets of minimal cardinality are investigated. In particular, if \(m(M)\) denotes the minimal cardinality of maximal equilateral sets in the metric space \(M\), the authors prove that \(m(C_0(X))\geq\omega\) whenever \(X\) is an infinite locally compact Hausdorff space (Theorem 3.5), with \(m(C(K))=\omega\) when \(K\) is an infinite compact metric space. Finally, they give several examples of non-metrizable compact spaces \(K\) such that \(m(C(K))=\omega\).