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 [Spaces with a regular $G_{\delta }$-diagonal, Topology Appl. 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 [Metrizability and the Fréchet-Urysohn property in topological groups, Proc. Amer. Math....

Let $T:[0,\infty )\to L\left(E\right)$ be a $C_0$-semigroup with unbounded generator $A:D\left(A\right)\to E$. We prove that $\left(T\right(t)x-x)/t$ has generically a very irregular behaviour for $x\notin D\left(A\right)$ as $t\to 0+$.