The 𝑄-topology, Whyburn type filters and the cluster set map

Abstract

We use nonstandard topology and the Q Q -topology to characterize normal, almost-normal, regular, almost-regular, semiregular spaces. The cluster [resp. θ \theta -cluster] set relation is used to characterize regular, almost-regular [resp. strongly-regular] spaces. The Whyburn [resp. Dickman] filter bases are characterized and it is shown that the cluster [resp. θ \theta -cluster] set relation restricted to the domain of the Whyburn [resp. Dickman] filter bases is an essentially continuous [resp. strongly θ \theta -continuous] map iff the space is Hausdorff [resp. Urysohn].