A note on functional tightness and minitightness of space of the $G$-permutation degree

We study the behavior of the minimal tightness and functional tightness of topological spaces under the influence of the functor of the permutation degree. Analytically: a) We introduce the notion of $\tau$-open sets and investigate some basic properties of them. b) We prove that if the map $f\colon X\rightarrow Y$ is $\tau$-continuous, then the map $SP^{n}f\colon SP^n X \rightarrow SP^n Y$ is also $\tau$-continuous. c) We show that the functor $SP^n$ preserves the functional tightness and the minimal tightness of compacts. d) Finally, we give some facts and properties on $\tau$-bounded spaces. More precisely, we prove that the functor of permutation degree $SP^n$ preserves the property of being $\tau$-bounded.