On the coincidence of some infimal hyperspace topologies of a metrizable space

Abstract

A metrizable topologycal space $X$ and the set $\Omega_X$ of all metrics generating the topology of $X$ are considered. It is well known that, in general, equivalent metrics $\rho$ and $\sigma$ from $\Omega_X$ can determine different Hausdorff metric topologies $\tau_{\widehat{\rho}}$ and $\tau_{\widehat{\sigma}}$, different proximal topologies $\tau_{\delta (\rho)}$ and $\tau_{\delta (\sigma)}$ and different Wijsman topologies $\tau_{W(\rho)}$ and $\tau_{W(\sigma)}$ on $\exp X$. Thus the families of topologies $\mathcal{T}_H = \{\tau_{\widehat{\rho}}\: | \: \rho\in\Omega_X\}$, $\mathcal{T}_{\delta} = \{\tau_{\delta (\rho)}\: | \: \rho\in\Omega_X\}$ and $\mathcal{T}_W = \{\tau_{W({\rho})}\: | \: \rho\in\Omega_X\}$ appear on the $\exp X$. We've described the cases when the infimums of this families, i.e. the topologies $\tau_{H(\inf)} = \inf\mathcal{T}_H$, $\tau_{\delta (\inf)} = \inf\mathcal{T}_{\delta}$ and $\tau_{W(\inf)} = \inf\mathcal{T}_W$ coincide: $\tau_{H(\inf)} =\tau_{\delta (\inf)}$ if and only if the space $X$ is second countable, $\tau_{H(\inf)} =\tau_{W(\inf)}$ and $\tau_{\delta (\inf)} = \tau_{W(\inf)}$ if and only if the space $X$ is compact. Besides it is found that the topologies $\tau_{\delta (\inf)}$ and $\tau_{W(\inf)}$ are sequential if and only if the space $X$ is second countable.