Some classes of topological spaces and the space of \(G\)-permutation degree. (English) Zbl 1539.18010
At the Prague Topological Symposium in 1981, V. V. Fedorchuk [Russ. Math. Surv. 36, No. 3, 211–233 (1981; Zbl 0495.54008); translation from Usp. Mat. Nauk 36, No. 3(219), 177–195 (1981)] posed the following general problem in the theory of covariant functors, which determined a new direction for research in the field of topology:
“Let \(\mathcal{P}\) be some topological (or geometric) property and let \(F\) be a covariant functor. If a topological space \(X\) has the property \(\mathcal{P}\), whether \(F(X)\) has the same property \(\mathcal{P}\)? Or vice versa, if \(F(X)\) has the property \(\mathcal P\), does it follow that the space \(X\) also has the property \(\mathcal{P}\)?”
In [L. D. R. Kočinac et al., Georgian Math. J. 31, No. 2, 285–291 (2024; Zbl 07829226)], the functor of \(G\)-permutation degree acting in the category of compact spaces and their continuous mappings was studied. It was proved that the functor \(SP_{G}^{n}\) preserves the network weight, \(\pi\)-character and local density of infinite topological \(T_1\)-spaces. The spaces of \(G\)-permutation degree were also investigated in [L. D. R. Kočinac et al., “Some cardinal and geometric properties of the space of permutation degree”, Axioms 11, No. 6, Paper No. 290 (2022); “Tightness-type properties of the space of permutation degree”, Mathematics 10, No. 18, Paper No. 3341 (2022)] and some tightness-type properties of the space \(SP_{G}^{n} X\) were considered. In [L. D. R. Kočinac et al., “Some cardinal and geometric properties of the space of permutation degree”, Axioms 11, No. 6, Paper No. 290 (2022)], it has been proved that \(SP_{G}^{n}\) is a covariant homotopy functor.
The authors of this paper, under the assumption that the topological spaces are \(T_1\)-space and \(F\) is the functor \(SP_{G}^{n}\), are dedicated to the investigation of the preservation of some kinds of topological spaces (such as \(r\)-spaces, cosmic spaces, \(C(k)\)-cosmic spaces, \(\alpha\)-spaces, and some other classes) under influence of the functor \(SP_{G}^{n}\).
Throughout the paper all spaces are assumed to be regular and \(k\) denotes an infinite cardinal number and the topology of a space \(X\) is denoted by \(\tau_{X}\).
“Let \(\mathcal{P}\) be some topological (or geometric) property and let \(F\) be a covariant functor. If a topological space \(X\) has the property \(\mathcal{P}\), whether \(F(X)\) has the same property \(\mathcal{P}\)? Or vice versa, if \(F(X)\) has the property \(\mathcal P\), does it follow that the space \(X\) also has the property \(\mathcal{P}\)?”
In [L. D. R. Kočinac et al., Georgian Math. J. 31, No. 2, 285–291 (2024; Zbl 07829226)], the functor of \(G\)-permutation degree acting in the category of compact spaces and their continuous mappings was studied. It was proved that the functor \(SP_{G}^{n}\) preserves the network weight, \(\pi\)-character and local density of infinite topological \(T_1\)-spaces. The spaces of \(G\)-permutation degree were also investigated in [L. D. R. Kočinac et al., “Some cardinal and geometric properties of the space of permutation degree”, Axioms 11, No. 6, Paper No. 290 (2022); “Tightness-type properties of the space of permutation degree”, Mathematics 10, No. 18, Paper No. 3341 (2022)] and some tightness-type properties of the space \(SP_{G}^{n} X\) were considered. In [L. D. R. Kočinac et al., “Some cardinal and geometric properties of the space of permutation degree”, Axioms 11, No. 6, Paper No. 290 (2022)], it has been proved that \(SP_{G}^{n}\) is a covariant homotopy functor.
The authors of this paper, under the assumption that the topological spaces are \(T_1\)-space and \(F\) is the functor \(SP_{G}^{n}\), are dedicated to the investigation of the preservation of some kinds of topological spaces (such as \(r\)-spaces, cosmic spaces, \(C(k)\)-cosmic spaces, \(\alpha\)-spaces, and some other classes) under influence of the functor \(SP_{G}^{n}\).
Throughout the paper all spaces are assumed to be regular and \(k\) denotes an infinite cardinal number and the topology of a space \(X\) is denoted by \(\tau_{X}\).
Reviewer: Joaquín Luna-Torres (Cartagena)
MSC:
| 18F60 | Categories of topological spaces and continuous mappings |
| 54B30 | Categorical methods in general topology |
| 54E99 | Topological spaces with richer structures |
Keywords:
functor of \(G\)-permutation degree; cosmic space; \(\alpha\)-space; \(r\)-space; \(\aleph_0\)-spaceCitations:
Zbl 0495.54008References:
| [1] | R. B. Beshimov, Some properties of the functor O_{\beta}, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 313 (2004), no. 11, 131-134; translation in J. Math. Sci. (N. Y.) 133 (2006), no. 5, 1599-1601. · Zbl 1073.18002 |
| [2] | R. B. Beshimov, D. N. Georgiou and N. K. Mamadaliev, On τ-bounded spaces and hyperspaces, Filomat 36 (2022), no. 1, 187-193. |
| [3] | R. B. Beshimov, D. N. Georgiou, F. Sereti and R. M. Zhuraev, Metric, stratifiable and uniform spaces of G-permutation degree, Math. Slovaca, to appear. · Zbl 07859883 |
| [4] | R. B. Beshimov, D. N. Georgiou and R. M. Zhuraev, Index boundedness and uniform connectedness of space of the G-permutation degree, Appl. Gen. Topol. 22 (2021), no. 2, 447-459. · Zbl 1514.54008 |
| [5] | R. B. Beshimov and D. T. Safarova, Some topological properties of a functor of finite degree, Lobachevskii J. Math. 42 (2021), no. 12, 2744-2753. · Zbl 1485.54013 |
| [6] | R. Engelking, General Topology, 2nd ed., Sigma Ser. Pure Math. 6, Heldermann, Berlin, 1989. · Zbl 0684.54001 |
| [7] | V. V. Fedorčuk, Covariant functors in a category of compacta, absolute retracts and Q-manifolds, Uspekhi Mat. Nauk 36 (1981), no. 3(219), 177-195. · Zbl 0495.54008 |
| [8] | V. V. Fedorchuk and V. V. Filippov, General Topology. The Basic Constructions (in Russian), Izd-vo MGU, Moscow, 1988. · Zbl 0658.54001 |
| [9] | V. V. Fedorchuk and V. V. Filippov, Topology of Hyperspaces and Its Applications, Current Life Sci. Technol. Ser. Math. Cybernetics 89, “Znanie”, Moscow, 1989. · Zbl 1081.54500 |
| [10] | D. N. Georgiou, S. D. Iliadis and A. C. Megaritis, C(\tau)-cosmic spaces, Topology Proc. 38 (2011), 149-164. · Zbl 1206.54024 |
| [11] | C. Good and S. Macías, Symmetric products of generalized metric spaces, Topology Appl. 206 (2016), 93-114. · Zbl 1342.54009 |
| [12] | K. P. Hart, J. Nagata and J. E. Vaughan, Encyclopedia of General Topology, Elsevier Science, Amsterdam, 2004. · Zbl 1059.54001 |
| [13] | L. D. R. Kočinac and F. G. Mukhamadiev, Some properties of the N^{\varphi}_{\tau}-nucleus, Topology Appl. 326 (2023), Paper No. 108430. · Zbl 1508.18001 |
| [14] | L. D. R. Kočinac, F. G. Mukhamadiev and A. K. Sadullaev, Some cardinal and geometric properties of the space of permutation degree, Axioms 11 (2022), no. 6, Paper No. 290. |
| [15] | L. D. R. Kočinac, F. G. Mukhamadiev and A. K. Sadullaev, Some topological and cardinal properties of the space of permutation degree, Filomat 36 (2022), no. 20, 7059-7066. |
| [16] | L. D. R. Kočinac, F. G. Mukhamadiev and A. K. Sadullaev, Tightness-type properties of the space of permutation degree, Mathematics 10 (2022), no. 18, Paper No. 3341. |
| [17] | E. Michael, \aleph_0-spaces, J. Math. Mech. 15 (1966), 983-1002. · Zbl 0148.16701 |
| [18] | F. G. Mukhamadiev, On certain cardinal properties of the N_{\tau}^{\varphi}-nucleus of a space X, J. Math. Sci. 245 (2020), no. 3, 411-415. · Zbl 1442.54006 |
| [19] | T. K. Yuldashev and F. G. Mukhamadiev, The local density and the local weak density in the space of permutation degree and in Hattori space, Ural Math. J. 6 (2020), no. 2, 108-116. · Zbl 1462.54010 |
| [20] | T. K. Yuldashev and F. G. Mukhamadiev, Caliber of space of subtle complete coupled systems, Lobachevskii J. Math. 42 (2021), no. 12, 3043-3047. · Zbl 1485.54008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
