π -Irreducible Mappings and K-Network of Infinite Compacts
R. B. Beshimov *
Department of Mathematics, Tashkent State Pedagogical University, Named After Nizami Str. Yusuf Khos Hojib 103, 100070 Tashkent, Uzbekistan.
N. K. Mamadaliev
Institute of Mathematics, National University of Uzbekistan, Named After Mirzo Ulugbek Durmon Yuli Str. 29, 100125 Tashkent, Uzbekistan.
F. G. Mukhamadiev
Department of Mathematics, Tashkent State Pedagogical University, Named After Nizami Str. Yusuf Khos Hojib 103, 100070 Tashkent, Uzbekistan.
*Author to whom correspondence should be addressed.
Abstract
In the paper the local density and the local weak density of topological spaces are investigated. It is proved that for a π-irreducible mapping f of a topological space X onto a topological space Y the followings hold: d(X) = d(Y ), wd(X) = wd(Y ), ld(X) ≤ ld(Y ), lwd(X) ≤ lwd(Y ). Moreover, it is showed that the functor of probability measures of nite supports Pn, the functor of the permutation degree and the functor expnpreserve the cardinality of k-networks of in nite compacts.
Keywords: π -irreducible mapping, k-network, the local density, the local weak density, hyperspace, the space of the permutation degree