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.2021.15566OJS447459222T. Banakh, Topological spaces with ith an ωω-base, Dissertationes Mathematicae, Warszawa, 2019. https://doi.org/10.4064/dm762-4-2018R. B. Beshimov, Nonincrease of density and weak density under weakly normal functors, Mathematical Notes 84 (2008), 493-497. https://doi.org/10.1134/S0001434608090216R. B. Beshimov, Some properties of the functor Oβ, Journal of Mathematical Sciences 133, no. 5 (2006), 1599-1601. https://doi.org/10.1007/s10958-006-0070-5R. 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.2019.106998R. 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.2017.02.037R. 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/s10958-020-04701-8R. 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.2019.070203A. 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)00120-0A. A. Borubaev and A. A. Chekeev, On uniform topology and its applications, TWMS J. Pure and Appl. Math. 6, no. 2 (2015), 165-179.R. Engelking, General topology, Berlin: Helderman, 1986.V. V. Fedorchuk, Covariant functors in the category of compacts, absolute ute retracts and Q-manifolds, Uspekhi Matematicheskikh Nauk 36, no. 3 (1981), 177-195. https://doi.org/10.1070/RM1981v036n03ABEH004251V. V. Fedorchuk and H. A. Kunzi, Uniformly open mappings and uniform embeddings of function spaces, Topology and its Applications 61 (1995), 61-84. https://doi.org/10.1016/0166-8641(94)00023-VV. V. Fedorchuk and V. V. Filippov, Topology of hyperspaces and its applications, 4 Mathematica, cybernetica, Moscow, 48 p., 1989 (in Russian).G. Itzkowitz, S. Rothman, H. Strassberg and T. S. Wu, Characterization of equivalent uniformities in topological groups, Topology and its Applications 47 (1992), 9-34. https://doi.org/10.1016/0166-8641(92)90112-DI. M. James, Introduction to Uniform Spaces, London Mathematical Society, Lecture Notes Series 144, Cambridge University Press, Cambridge, 1990.J. L. Kelley, General Topology, Van Nostrand Reinhold, Princeton, NJ, 1955.L. Holá and L. D. R. Kocinac, Uniform boundedness in function spaces, Topology and its Applications 241 (2018), 242-251. https://doi.org/10.1016/j.topol.2018.04.006E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. https://doi.org/10.1090/S0002-9947-1951-0042109-4T. N. Radul, On the functor of order-preserving functionals, Comment. Math. Univ. Carol. 39, no. 3 (1998), 609-615.T. K. Yuldashev and F. G. Mukhamadiev, The local density and the local weak density in the space of permutation degree and in Hattorri space, URAL Mathematical Journal 6, no. 2 (2020), 108-126. https://doi.org/10.15826/umj.2020.2.01

SOME PROPERTIES OF TOPOLOGICAL SPACES RELATED TO THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY

Abstract: In the paper the local density and the local weak density of topological spaces are investigated. It is proved that for stratifiable spaces the local density and the local weak density coincide, these cardinal numbers are preserved under open mappings, are inverse invariant of a class of closed irreducible mappings. Moreover, it is showed that the functor of probability measures of finite supports preserves the local density of compacts