Notes on paratopological groups

In this paper, we prove that if a remainder of a non-locally compact paratopological group G   has a _Gδ$G_{\delta }$-diagonal and every compact subset of G is first countable, then G   has a _Gδ$G_{\delta }$-diagonal of infinite rank. This improves a result of Chuan Liu and Shou Lin Chuan Liu, Shou Lin, Generalized metric spaces with algebraic structure, Topology Appl. 157 (2010) 1966–1974]. We also construct an open continuous homomorphism f from a non-metrizable paratopological group G onto a metrizable topological group H such that the kernel of f is metrizable. This result gives a negative answer to an open problem posed in A.V. Arhangel?skii, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, World Scientific, 2008].