### THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE

#### Abstract

In this paper, the local density \((l d)\) and the local weak density \((l w d)\) in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree \(S P^{n}\) and the subfunctor of permutation degree \(S P_{G}^{n}\), \(P\) is the cardinal number of topological spaces. Let \(X\) be an infinite \(T_{1}\)-space. We prove that the following propositions hold.

(1) Let \(Y^{n} \subset X^{n}\); (A) if \(d\, \left(Y^{n} \right)=d\, \left(X^{n} \right)\), then \(d\, \left(S P^{n} Y\right)=d\, \left(SP^{n} X\right)\); (B) if \(l w d\, \left(Y^{n} \right)=l w d\, \left(X^{n} \right)\), then \(l w d\, \left(S P^{n} Y\right)=l w d\, \left(S P^{n} X\right)\).

(2) Let \(Y\subset X\); (A) if \(l d \,(Y)=l d \,(X)\), then \(l d\, \left(S P^{n} Y\right)=l d\, \left(S P^{n} X\right)\); (B) if \(w d \,(Y)=w d \,(X)\), then \(w d\, \left(S P^{n} Y\right)=w d\, \left(S P^{n} X\right)\).

(3) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is a locally compact \(T_{1}\)-space, then \(S P^{n} X, \, S P_{G}^{n} X\), and \(\exp _{n} X\) are \(k\)-spaces.

(4) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is an infinite \(T_{1}\)-space, then \(n \,\pi \,w \left(X\right)=n \, \pi \,w \left(S P^{n} X \right)=n \,\pi \,w \left(S P_{G}^{n} X \right)=n \,\pi \,w \left(\exp _{n} X \right)\).

We also have studied that the functors \(SP^{n},\) \(SP_{G}^{n} ,\) and \(\exp _{n} \) preserve any \(k\)-space. The functors \(SP^{2}\) and \(SP_{G}^{3}\) do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite \(T_{1}\)-space \(X\) coincides with the densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\). It is also shown that the weak density of an infinite \(T_{1}\)-space \(X\) coincides with the weak densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\).

#### Keywords

#### Full Text:

PDF#### References

Aleksandrov P.S., Fedorchuk V.V., Zaitsev V.I. The main aspects in the development of set-theoretical topology. *Russian Math. Surveys*, 1978. Vol. 33, No. 3. P. 1–53. DOI: 10.1070/RM1978v033n03ABEH002464

Beshimov R.B., Mamadaliev N.K., Mukhamadiev F.G. Some properties of topological spaces related to the local density and the local weak density. *Math. Stat.*, 2015. Vol. 3, No. 4. P. 101–105. DOI: 10.13189/ms.2015.030404

Beshimov R.B. A note on weakly separable spaces. *Math. Morav.*, 2002. Vol. 6. P. 9–19. DOI: 10.5937/MatMor0206009B

Beshimov R.B. Some cardinal properties of topological spaces connected with weakly density. *Methods Funct. Anal. Topology*, 2004. Vol. 10, No. 3. P. 17–22. DOI: http://mfat.imath.kiev.ua/article/?id=251

Beshimov R.B., Mukhamadiev F.G. Cardinal properties of Hattori spaces and their hyperspaces. *Questions Answers Gen. Topology*, 2015. Vol. 33, No. 1. P. 43–48.

Beshimov R.B., Mukhamadiev F.G., Mamadaliev N.K. The local density and the local weak density of hyperspaces. *Int. J. Geom.*, 2015. Vol. 4, No. 1. P. 42–49.

Engelking R. *General Topology*. Berlin: Heldermann Verlag, 1989. 529 p.

Fedorchuk V.V. Covariant functors in the category of compacts, absolute retracts, and Q-manifolds. *Russian Math. Surveys*, 1981. Vol. 36, No. 3. P. 211–233. DOI: 10.1070/RM1981v036n03ABEH004251

Fedorchuk V.V., Filippov V.V. *Topology of Hyperspaces and its Applications*. Moscow: Mathematica, Cybernetica, 1989. Vol. 4. 48 p. (in Russian)

Hattori Y. Order and topological structures of posets of the formal balls on metric spaces. *Mem. Fac. Sci. Eng. Shimane Univ. Ser. B: Math. Sci.*, 2010. Vol. 43. P. 13–26.

Michael E. Topologies on spaces of subsets. *Trans. Amer. Math. Soc.*, 1951. Vol. 71, No. 1. P. 152–182. DOI: 10.1090/S0002-9947-1951-0042109-4

Mukhamadiev F.G. Some cardinal and topological properties of the n-permutation degree of a topological spaces and locally \(\tau\)-density of hyperspaces. *Bull. Nat. Univ. Uzbekistan: Math. Nat. Sci.*, 2018. Vol. 1, No. 1. Art. no. 11. P. 30–35 https://uzjournals.edu.uz/mns_nuu/vol1/iss1/11

Wagner C.H. *Symmetric, Cyclic, and Permutation Products of Manifolds.* Warszawa: PWN, 1980. 48 p.

#### Article Metrics

### Refbacks

- There are currently no refbacks.