Topologies larger than \(\alpha\)-compact topologies (Q1110844)

Denote by \(m(\alpha)\) the first measurable cardinal for a topology T let \(T(\kappa)\) be the \(G_{\kappa}\)-modification of the topology T, \(\kappa\) be an infinite cardinal. Generalizing results of W. W. Comfort, T. Retta, Z. Frolík, S. W. Williams and others, the authors prove that if (X,T) is an \(\alpha\)-compact space in the sense of Herrlich and T' is a Tikhonov topology such that \(T\subset T'\subset T(\kappa)\), then (X,T') is \(\alpha\)-compact in each of the following cases: (1) \(\kappa\leq \alpha\), (2) (X,T') is a \(P_{\kappa}\)-space, (3) \(\kappa =\mu^+\) and \(T(\mu)\subset T'\), (4) \(T'=T(\kappa)\). The last part of the paper deals with \(\alpha\)-compactness of box products. It is shown that if \(\omega\leq \kappa \leq m(\alpha)\) then any \(\kappa\)-box product of \(\alpha\)-compact zero-dimensional spaces is \(\alpha\)-compact and every closed discrete subspace of a \(\kappa\)-box product of \(\alpha\)-compact spaces is \(\alpha\)-compact.