Exercise 2, § 3

The definition of the left and right uniformities on a topological group $G$ uses only the properties $(\mathrm{GV}_I)$ and $(\mathrm{GV}_{II})$ of the neighbourhood filter $\mathfrak{B}$ of $e$ in $G$. If we suppose that $\mathfrak{B}$ is a filter satisfying these two axioms, but not necessarily satisfying axiom $(\mathrm{GV}_{III})$, show that the left and right uniformities are respectively compatible with the topologies $\mathscr{C}_s$ and $\mathscr{C}_d$ defined in Exercise 5 a) of § 1.