When the finest splitting topology is a group topology or Fréchet (Q386213)

The splitting topology (under the name proper topology) in function spaces was introduced by \textit{R. Arens} and \textit{J. Dugundji} [Pac. J. Math. 1, 5--31 (1951; Zbl 0044.11801)]. The author investigates the space \(C_{T(c)}(X)\) of real-valued continuous functions on a completely regular space \(X\) endowed with the finest splitting topology \(T(c)\) and obtains a number of interesting results. It is proved that Lindelöfness of \(X\) is equivalent to each of the following properties of the space \(C_{T(c)}(X)\): (1) \(C_{T(c)}(X)\) is sequential, (2) \(C_{T(c)}(X)\) is a \(k\)-space, (3) \(C_{T(c)}(X)\) is countably tight. For a special class of Lindelöf spaces, called sequentially inaccessible, \(C_{T(c)}(X)\) is a Fréchet space if and only if it is sequential and \(\alpha_2\) if and only if it is sequential and \(\alpha_4\). It is also proved that for a Lindelöf space \(X\), \(C_{T(c)}(X)\) is a Fréchet topological group if and only if its countable power has the same property.