Recognizing special metrics by topological properties of the ``metri''-proximal hyperspace

Abstract

In this paper, we first characterize those compatible metrics $d$ on a metrizable space $X$ which give rise to a connected $d$-proximal hyperspace. We show that the space of irrational numbers, in particular, admits a complete metric with this property and, as a consequence, we get a negative answer to a question of [11] about selections for hyperspace topologies. Next, we characterize the compatible metrics on $X$ which are uniformly equivalent to ultrametrics showing that this is equivalent to the zero-dimensionality of the corresponding proximal hyperspaces. Applications and related results about other disconnectedness-like properties of proximal hyperspaces are obtained.