In this paper, we discuss certain networks on paratopological (or topological) groups and give positive or negative answers to the questions in [Lin2013]. We also prove that a non-locally compact, $k$-gentle paratopological group is metrizable if its remainder (in the Hausdorff compactification) is a Fréchet-Urysohn space with a point-countable cs*-network, which improves some theorems in [Liu C., Metrizability of paratopological $($semitopological$)$ groups, Topology Appl. 159 (2012), 1415–1420], [Liu...

The following general question is considered. Suppose that $G$ is a topological group, and $F$, $M$ are subspaces of $G$ such that $G=MF$. Under these general assumptions, how are the properties of $F$ and $M$ related to the properties of $G$? For example, it is observed that if $M$ is closed metrizable and $F$ is compact, then $G$ is a paracompact $p$-space. Furthermore, if $M$ is closed and first countable, $F$ is a first countable compactum, and $FM=G$, then $G$ is also metrizable. Several other results of this kind are obtained....

Every nontrivial countably productive coreflective subcategory of topological linear spaces is $\kappa$-productive for a large cardinal $\kappa$ (see [10]). Unlike that case, in uniform spaces for every infinite regular cardinal $\kappa$, there are coreflective subcategories that are $\kappa$-productive and not ${\kappa }^{+}$-productive (see [8]). From certain points of view, the category of topological groups lies in between those categories above and we shall show that the corresponding results on productivity of coreflective subcategories...