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

Herrmann, Robert A.

doi:10.1090/S0002-9939-1976-0482730-5

The $Q$-topology, Whyburn type filters and the cluster set map
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Proc. Amer. Math. Soc. 59 (1976), 161-166 Request permission

Erratum: Proc. Amer. Math. Soc. 65 (1977), 375.

Abstract:

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

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