## Dimitrios N. Georgiou, Nodirbek K. Mamadaliev, Rustam M. Zhuraev

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

Comment.Math.Univ.Carolin. 64,1 (2023) 97-108.**Abstract: **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.

**Keywords:** \tau-open set; \tau-bounded space; functional tightness; minimal tightness

**DOI:** DOI 10.14712/1213-7243.2023.019

**AMS Subject Classification:** 54C05 54B20

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