An Ascoli theorem for sequential spaces (Q5954787)

In this nice paper the author gives a transition from the Ascoli-theory for topological spaces to that of sequential spaces. The importance of this kind of investigation becomes clear from a practical point of view: practicians mostly work with sequences instead of filters or nets -- and they are interested in results concerning sequences, even if the corresponding results would not hold for general nets or filters. Moreover, it is an interesting variation to investigate topological categories (even cartesian closed ones) based on the countable concept of sequences.NEWLINENEWLINENEWLINEUnfortunately, the paper contains a gap, namely the proposition 9.1, stating that a set of continuous functions from a sequential space with a limit-space-like property to a sequential space with Uryson property would be evenly continuous, whenever it is relatively compact in the set of all continuous functions with respect to continuous convergence, and the range space is \(R_0\). The \(R_0\) property is not sufficient for the essential step in the author's argumentation, and the proposition is consistently false, as can be seen by examples. Uniformizability (as defined here by the author) instead of \(R_0\) would work fine. For this reason, the author's Ascoli-theorems remain still true. An interesting application to separable \(C^*\)-algebras is given.