A note on paratopological groups

summary:In this paper, it is proved that a first-countable paratopological group has a regular $G_{\delta}$-diagonal, which gives an affirmative answer to Arhangel'skii and Burke's question [{\it Spaces with a regular $G_{\delta}$-diagonal\/}, Topology Appl. {\bf 153} (2006), 1917--1929]. If $G$ is a symmetrizable paratopological group, then $G$ is a developable space. We also discuss copies of $S_\omega$ and of $S_2$ in paratopological groups and generalize some Nyikos [{\it Metrizability and the Fréchet-Urysohn property in topological groups\/}, Proc. Amer. Math. Soc. {\bf 83} (1981), no. 4, 793--801] and Svetlichnyi [{\it Intersection of topologies and metrizability in topological groups\/}, Vestnik Moskov. Univ. Ser. I Mat. Mekh. {\bf 4} (1989), 79--81] results