A NOTE ON -SPACES

In this paper, it is shown that every compact Hausdorff $K$-space has countable tightness. This result gives a positive answer to a problem posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl.104 (2000), 181–190]. We show that a semitopological group $G$ that is a $K$-space is first countable if and only if $G$ is of point-countable type. It is proved that if a topological group $G$ is a $K$-space and has a locally paracompact remainder in some Hausdorff compactification, then $G$ is metrisable.

TheD-Property of Finite Unions oft-Metrizable Spaces and Certain Function Spaces

The additivity ofD-property is studied ont-metrizable spaces and certain function spaces. It is shown that a space of countable tightness is aD-space provided that it is the union of finitely manyt-metrizable subspaces, or function spacesCp(Xi)where eachXiis LindelöfΣ.