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arXiv · 1806.11341

A parallel metrization theorem

Taras Banakh·Olena Hryniv
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Abstract

Two non-empty sets $A,B$ of a metric space $(X,d)$ are called parallel if $d(a,B)=d(A,B)=d(A,b)$ for any points $a\in A$ and $b\in B$. Answering a question posed on Mathoverflow, we prove that for a cover $\mathcal C$ of a metrizable space $X$ the following conditions are equivalent: (i) the topology of $X$ is generated by a metric $d$ such that any two sets $A,B\in\mathcal C$ are parallel; (ii) the cover $\mathcal C$ is disjoint, lower semicontinuous and upper semicontinuous.

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Taras Banakh, Olena Hryniv. 2018-06-29. A parallel metrization theorem. https://doi.org/10.1007/s40879-018-00311-6

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