A few generalized metric properties on free paratopological groups

In this paper, some generalized metric properties on free paratopological groups are obtained. At first, a general stability theorem of free paratopological groups on submetrizable spaces is established. Secondly, countable tightness of free paratopological groups on k⁎k⁎-metrizable k-spaces or k-semistratifiable k-spaces is investigated. Finally, first-countability of free paratopological groups is discussed and strongly Fréchetness of free paratopological groups is characterized. These complement and improve some known conclusions in literature. In addition, some questions are posed.