The countable type properties in free paratopological groups

Abstract: A space $X$ is of countable type (resp. subcountable type) if every compact subspace $F$ of $X$ is contained in a compact subspace $K$ that is of countable character (resp. countable pseudocharacter) in $X$. In this paper, we mainly show that: (1) For a functionally Hausdorff space $X$, the free paratopological group $FP(X)$ and the free abelian paratopological group $AP(X)$ are of countable type if and only if $X$ is discrete; (2) For a functionally Hausdorff space $X$, if the free abelian paratopological group $AP(X)$ is of subcountable type then $X$ has countable pseudocharacter. Moreover, we also show that, for an arbitrary Hausdorff $\mu$-space $X$, if $AP_{2}(X)$ or $FP_{2}(X)$ is locally compact, then $X$ is homeomorphic to the topological sum of a compact space and a discrete space.