Abstract: Abstract 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.