Geodesic
A note on paratopological groups
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 4, pp. 633-640.

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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.
Classification : 22A30, 54E30, 54H13, 54H99
Keywords: paratopological group; symmetrizable spaces; regular $G_{\delta}$-diagonal; weak bases; Arens space 
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Liu, Chuan. A note on paratopological groups. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 4, pp. 633-640. http://geodesic.mathdoc.fr/item/CMUC_2006__47_4_a8/