The Weak Density and the Local Weak Density of Space of the Permutation Degree and Hyperspaces

Mukhamadiev, F. G.; Sadullaev, Anvar K.; Meyliev, Sh. U.

doi:10.37418/amsj.9.12.28

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“…For every mapping f : X → Y, the mapping [1,4,6]. The weak density of a space X, denoted by wd(X), is the smallest cardinal number τ ≥ ω such that there is a π-base B = ∪{B α : α < τ} in X, and for each α < τ, B α is a centered system of open sets in X [7,9,10].…”

Section: Preliminariesmentioning

confidence: 99%

“…In recent research, the interest in the theory of cardinal invariants and their behavior under the influence of various covariant functors is increasing (see, for example, [7][8][9]). In [10], the authors investigated several cardinal invariants under the influence of some seminormal and normal functors. In the investigations in [11,12], the concept of symmetric product of a topological space is introduced.…”

Section: Introductionmentioning

confidence: 99%

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Some Cardinal and Geometric Properties of the Space of Permutation Degree

Kočinac

Mukhamadiev

Sadullaev

2022

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This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP2(R,τ(A)) of permutation degree of the Hattori space is discussed. In particular, it is shown that the space SP2(R,τS) contains a closed discrete subset of cardinality c. Moreover, it is shown that the functor SPGn preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces SPGnX and SPGnY are also homotopically equivalent. As a result, it has been proved that the functor SPGn is a covariant homotopy functor.

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