arXiv subjects
Publications and source records attributed to Ruslan Shanin.
Let $X$ be a set and $2^X$ be a set of all subsets of $X$. The necessary and sufficient conditions under which a mapping $X \to 2^X$ is a closure of one-point sets in some $T_0$-space $(X, τ)$ are described. It is proved that every $T_0$-Alexandroff space is quasi-metrizable by some equidistant quasi-metric.
For arbitrary semimetric space $(X, d)$ and disjoint proximinal subsets $A$, $B$ of $X$ we define the proximinal graph as a bipartite graph with parts $A$ and $B$ whose edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A, B)$. We characterize the semimetric spaces whose proximinal graphs have at most one edge and the semimetric spaces whose proximinal graphs have the vertices with degree at most $1$ only. This allows us to describe the necessary and sufficient conditions for uniqueness of the best proximity pairs and best approximations.
Let $WS(X, d)$ be the class of ultrametric spaces which are weakly similar to ultrametric space $(X, d)$. The main results of the paper completely describe the ultrametric spaces $(X, d)$ for which the equality $$ ρ(x, y) = f(d(Φ(x), Φ(y))) $$ holds for every $(Y, ρ) \in WS(X, d)$, every weak similarity $Φ\colon Y \to X$, and all $x$, $y \in Y$ with some ultrametric (pseudoultrametric) preserving function $f$ depending on $Φ$.