Uncountable sets of unit vectors that are separated by more than \(1\) (Q2804307)

This paper provides a substantial improvement of the well-known results by \textit{C. A. Kottman} [Stud. Math. 53, 15--27 (1975; Zbl 0266.46014)] and \textit{J. Elton} and \textit{E. Odell} [Colloq. Math. 44, 105--109 (1981; Zbl 0493.46014)] concerning the existence of countable sets separated by more than \(1\) in the unit sphere of infinite-dimensional Banach spaces.NEWLINENEWLINEIn fact, the authors list many classes of spaces with high density character, the unit sphere of them containing uncountable sets separated by more than \(1\). In particular, they show that this is the case for Banach spaces containing non-separable reflexive subspaces (so in particular for non-separable quasi-reflexive spaces) as well as for the weakly Lindelöf determined Banach spaces (the weakly compactly generated spaces are among them) of density character greater than that of the continuum. Moreover, existence is guaranteed in the unit sphere of super-reflexive spaces, of uniformly separated sets by more than \(1\) of cardinality \(k\), where \(k\) is any regular cardinal not exceeding the density character of the space (so, in particular, uncountable \((1+\epsilon)\)-separated sets for some \(\epsilon > 0\) lie in the unit sphere of any non-separable super-reflexive space). Finally, it is proved that the unit sphere of the space \(C(K)\), \(K\) any non-metrizable compact Hausdorff space, contains an uncountable \(2\)-equilateral set whenever \(K\) is not perfectly normal, while it contains a set separated by more than \(1\) of the same cardinality as the density character of \(C(K)\) if \(K\) is perfectly normal, thus solving a problem left open by \textit{S. K. Mercourakis} and \textit{G. Vassiliadis} [Stud. Math. 231, No. 3, 241--255 (2015; Zbl 1361.46015)].