### A counterexample in semimetric spaces.

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Following the ideas of R. DeMarr, we establish a Galois connection between distance functions on a set $S$ and inequality relations on ${X}_{S}=S\times \mathbb{R}$. Moreover, we also investigate a relationship between the functions of $S$ and ${X}_{S}$.

An existing description of the cartesian closed topological hull of $p{\mathbf{\text{MET}}}^{\infty}$, the category of extended pseudo-metric spaces and nonexpansive maps, is simplified, and as a result, this hull is shown to be a special instance of a “family” of cartesian closed topological subconstructs of $pqs{\mathbf{\text{MET}}}^{\infty}$, the category of extended pseudo-quasi-semi-metric spaces (also known as quasi-distance spaces) and nonexpansive maps. Furthermore, another special instance of this family yields the cartesian closed topological hull of...

In this paper, we prove that a space $X$ is a $g$-metrizable space if and only if $X$ is a weak-open, $\pi $ and $\sigma $-image of a semi-metric space, if and only if $X$ is a strong sequence-covering, quotient, $\pi $ and $mssc$-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”.

In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We...