For a Tychonoff space $X$, let $\downarrow {\mathrm{C}}_F\left(X\right)$ be the family of hypographs of all continuous maps from $X$ to $[0,1]$ endowed with the Fell topology. It is proved that $X$ has a dense separable metrizable locally compact open subset if $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable. Moreover, for a first-countable space $X$, $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable if and only if $X$ itself is a locally compact separable metrizable space. There exists a Tychonoff space $X$ such that $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable but $X$ is not first-countable.

We consider the class of compact spaces $K_A$ which are modifications of the well known double arrow space. The space $K_A$ is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of $K_A$ spaces and on the isomorphic classification of the Banach spaces $C\left(K_A\right)$.

We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $\mu G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $\mu G$. This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...