On $\tau$-bounded spaces and hyperspaces

Abstract

In this paper, we prove facts and properties on τ -bounded spaces, which are introduced in [1]. More precisely, we prove that an arbitrary product of $\tau$ -bounded spaces is $\tau$ bounded and vise versa, and that the $\tau$ -bounded property is preserved by $\tau$-continuous maps. In particular, continuous maps preserve $\tau$-bounded spaces. Moreover, we investigate the behavior of the minimal tightness and functional tightness of topological spaces under the influence of an exponential functor of finite degree. It is proved that this functor preserves the functional tightness and the minimal tightness of compact sets.