The $\pi$-weights and $\pi$-characters of hyperspaces with the hit-and-miss topologies

Abstract

We prove that $\pi w\left(\Delta\left(x\right)\right)=\pi\chi\left(\Delta\left(x\right)\right)=\textrm{max}\left\{ \pi w\left(x\right),\tau k\left(\Delta\right)\right\} $ for the hit-and-miss topologies $\tau\Delta$ on the closed subsets of either a quasi-regular and $\mathbf{R}_{0}$ or $T_{1}$ space $\left(X,\tau\right)$, where $\tau k\left(\Delta\right)$ is a cardinal invariant associted with $\Delta$.