On functional tightness and minitightness of the exponential space

In this talk, we study the behavior of the minimal tightness and functional tightness of topological spaces under the influence of the exponential functor of finite degree. It is proved that the functor preserves the functional tightness and the minimal tightness of compact sets. And also, $\tau$-continuity of the mapping $\exp _n f\colon \exp _n (X) \rightarrow \exp _n (Y)$ is proved for any $\tau$-continuous mapping. Besides, we shall prove some facts and properties on $\tau$-bounded spaces. More precisely, we prove that an arbitrary product of $\tau$-bounded spaces is $\tau$-bounded and vise versa, $\tau$-boundedness is preserved by $\tau$-continuous maps, in particular, continuous maps preserve $\tau$-bounded spaces.