Index boundedness and uniform connectedness of space of the G-permutation degree

[EN] In this paper the properties of space of the G-permutation degree, like: weight, uniform connectedness and index boundedness are studied. It is proved that: (1) If (X, U) is a uniform space, then the mapping π s n, G : (X n , U n ) → (SP n GX, SP n GU) is uniformly continuous and uniformly open, moreover w (U) = w (SP n GU); (2) If the mapping f : (X, U) → (Y, V) is a uniformly continuous (open), then the mapping SP n Gf : (SP n GX, SP n GU) → (SP n GY, SP n GV) is also uniformly continuous (open); (3) If the uniform space (X, U) is uniformly connected, then the uniform space (SP n GX, SP n GU) is also uniformly connected. Beshimov, RB.; Georgiou, DN.; Zhuraev, RM. (2021). Index boundedness and uniform connectedness of space of the G-permutation degree. Applied General Topology. 22(2):447-459. https://doi.org/10.4995/agt.2021.15566 T. Banakh, Topological spaces with ith an ωω-base, Dissertationes Mathematicae, Warszawa, 2019. https://doi.org/10.4064/dm762-4-2018 R. B. Beshimov, Nonincrease of density and weak density under weakly normal functors, Mathematical Notes 84 (2008), 493-497. https://doi.org/10.1134/S0001434608090216 R. B. Beshimov, Some properties of the functor Oβ, Journal of Mathematical Sciences 133, no. 5 (2006), 1599-1601. https://doi.org/10.1007/s10958-006-0070-5 R. B. Beshimov and N. K. Mamadaliev, Categorical and topological properties of the functor of Radon functionals, Topology and its Applications 275 (2020), 1-11. https://doi.org/10.1016/j.topol.2019.106998 R. B. Beshimov and N. K. Mamadaliev, On the functor of semiadditive τ-smooth functionals, Topology and its Applications 221, no. 3 (2017), 167-177. https://doi.org/10.1016/j.topol.2017.02.037 R. B. Beshimov, N. K. Mamadaliev, Sh. Kh. Eshtemirova, Categorical and cardinal properties of hyperspaces with a finite number of components, Journal of Mathematical Sciences 245, no. 3 (2020), 390-397. https://doi.org/10.1007/s10958-020-04701-8 R. B. Beshimov and R. M. Zhuraev, Some properties of a connected topological group, Mathematics and Statistics 7, no. 2 (2019), 45-49. https://doi.org/10.13189/ms.2019.070203 A. A. Borubaev and A. A. Chekeev, On completions of topological groups with respect to the maximal uniform structure and factorization of uniform homomorphisms with respect to uniform weight and dimension, Topology and its Applications 107, no. 1-2 (2000), 25-37. https://doi.org/10.1016/S0166-8641(99)00120-0 A. A. Borubaev and A. A. Chekeev, On uniform topology and its appli