Abstract
In this paper, we study several generalized metric properties of the space $\mathcal F(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. In particular, we consider such properties $(P)$ for which $\mathcal F(X)$ has $(P)$ if and only if $X$ has $(P)$. Also, we obtain some results related to the images of metric spaces under some kinds of continuous mappings.
Keywords
hyperspace generalized metric properties $cs$-cover $cs^*$-cover $sn$-symmetric space (weak) Cauchy symmetric space (weak) Cauchy $sn$-symmetric space
References
- [1] T.V. An and L.Q. Tuyen, On an affirmative answer to S. Lin’s problem, Topology Appl. 158, 1567–1570, 2011.
- [2] T.V. An and L.Q. Tuyen, Cauchy $sn$-symmetric spaces with a $cs$-network ($cs^*$-network) having property $\sigma$-$(P)$}, Topology Proc. 51, 61–75, 2018.
- [3] T.V. An and L.Q. Tuyen, Spaces with sn-network g-functions, Topology Proc. 54, 177–191, 2019.
- [4] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
- [5] X. Ge, Mappings on weak Cauchy sn-symmetric spaces, Lobachevskii J. Math. 30 (2), 132–137, 2009.
- [6] X. Ge and J. Li, Characterizations of weak Cauchy sn-symmetric spaces, Gen. Math. 18 (2), 3–13, 2010.
- [7] Y. Ge and S. Lin, g-metrizable spaces and the images of semi-metric spaces, Czech. Math. J. 57(132), 1141–1149, 2007.
- [8] C. Good and S. Macías, Symmetric products of generalized metric spaces, Topology Appl. 206, 93–114, 2016.
- [9] Y. Ikeda, C. Liu and Y. Tanaka, Quotient compact images of metric spaces, and related matters, Topology Appl. 122, 237–252, 2002.
- [10] F. Lin, R. Shen and C. Liu, Generalized metric properties on hyperspaces with the Vietoris topology, Rocky Mt. J. Math. 51 (5), 1761–1779, 2021.
- [11] S. Lin and P. Yan, On sequence-covering compact mappings, Acta Math. Sinica 44, 175–182, 2001.
- [12] S. Lin and P. Yan, Sequence-covering maps of metric spaces, Topology Appl. 109, 301–314, 2001.
- [13] S. Lin and Z. Yun,Generalized Metric Spaces and Mappings, Atlantis Studies in Mathematics, vol. 6, Atlantis Press, Paris, 2016.
- [14] X. Liu, C. Liu and S. Lin, Strict Pytkeev networks with sensors and their applications in topological groups, Topology Appl. 258, 58–78, 2019.
- [15] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71, 91–138, 1951.
- [16] L. Mou, P. Li and S. Lin, Regular $G_\delta$-diagonals and hyperspaces, Topology Appl. 301, 107530, 2021.
- [17] L.-X. Peng and Y. Sun, A study on symmetric products of generalized metric spaces, Topology Appl. 231, 411–429, 2017.
- [18] Y. Tanaka and Y. Ge, Around quotient compact images of metric spaces, and symmetric spaces, Houston J. Math. 32 (1), 99–117, 2006.
- [19] Y. Tanaka and Z. Li, Certain covering-maps and k-networks, and related matters, Topology Proc. 27 (1), 317–334, 2003.
- [20] Z. Tang, S. Lin and F. Lin, Symmetric products and closed finite-to-one mappings, Topology Appl. 234, 26–45, 2018.
- [21] L.Q. Tuyen and O.V. Tuyen, On the $n$-fold symmetric product of a space with a $\sigma$-$(P)$-property $cn$-network ($ck$-network), Comment. Math. Univ. Carolinae 61 (2), 257–263, 2020.
- [22] L.Q. Tuyen and O.V. Tuyen, A note on the hyperspace of finite subsets, Fasciculi Math. 65, 67–73, 2021.
- [23] N.V. Velichko, Symmetrizable spaces, Math. Notes 12, 784–786, 1972.
- [24] W.A. Wilson, On semi-metric spaces, Amer. J. Math. 53, 361–373, 1931.
- [25] P.F. Yan, On strong sequence-covering compact mappings, Northeast. Math. J. 14 (3), 341–344, 1998.
Abstract
Keywords
Hyperspace generalized metric properties $cs$-cover $cs^*$-cover $sn$-symmetric space (weak) Cauchy symmetric space (weak) Cauchy $sn$-symmetric space
References
- [1] T.V. An and L.Q. Tuyen, On an affirmative answer to S. Lin’s problem, Topology Appl. 158, 1567–1570, 2011.
- [2] T.V. An and L.Q. Tuyen, Cauchy $sn$-symmetric spaces with a $cs$-network ($cs^*$-network) having property $\sigma$-$(P)$}, Topology Proc. 51, 61–75, 2018.
- [3] T.V. An and L.Q. Tuyen, Spaces with sn-network g-functions, Topology Proc. 54, 177–191, 2019.
- [4] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
- [5] X. Ge, Mappings on weak Cauchy sn-symmetric spaces, Lobachevskii J. Math. 30 (2), 132–137, 2009.
- [6] X. Ge and J. Li, Characterizations of weak Cauchy sn-symmetric spaces, Gen. Math. 18 (2), 3–13, 2010.
- [7] Y. Ge and S. Lin, g-metrizable spaces and the images of semi-metric spaces, Czech. Math. J. 57(132), 1141–1149, 2007.
- [8] C. Good and S. Macías, Symmetric products of generalized metric spaces, Topology Appl. 206, 93–114, 2016.
- [9] Y. Ikeda, C. Liu and Y. Tanaka, Quotient compact images of metric spaces, and related matters, Topology Appl. 122, 237–252, 2002.
- [10] F. Lin, R. Shen and C. Liu, Generalized metric properties on hyperspaces with the Vietoris topology, Rocky Mt. J. Math. 51 (5), 1761–1779, 2021.
- [11] S. Lin and P. Yan, On sequence-covering compact mappings, Acta Math. Sinica 44, 175–182, 2001.
- [12] S. Lin and P. Yan, Sequence-covering maps of metric spaces, Topology Appl. 109, 301–314, 2001.
- [13] S. Lin and Z. Yun,Generalized Metric Spaces and Mappings, Atlantis Studies in Mathematics, vol. 6, Atlantis Press, Paris, 2016.
- [14] X. Liu, C. Liu and S. Lin, Strict Pytkeev networks with sensors and their applications in topological groups, Topology Appl. 258, 58–78, 2019.
- [15] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71, 91–138, 1951.
- [16] L. Mou, P. Li and S. Lin, Regular $G_\delta$-diagonals and hyperspaces, Topology Appl. 301, 107530, 2021.
- [17] L.-X. Peng and Y. Sun, A study on symmetric products of generalized metric spaces, Topology Appl. 231, 411–429, 2017.
- [18] Y. Tanaka and Y. Ge, Around quotient compact images of metric spaces, and symmetric spaces, Houston J. Math. 32 (1), 99–117, 2006.
- [19] Y. Tanaka and Z. Li, Certain covering-maps and k-networks, and related matters, Topology Proc. 27 (1), 317–334, 2003.
- [20] Z. Tang, S. Lin and F. Lin, Symmetric products and closed finite-to-one mappings, Topology Appl. 234, 26–45, 2018.
- [21] L.Q. Tuyen and O.V. Tuyen, On the $n$-fold symmetric product of a space with a $\sigma$-$(P)$-property $cn$-network ($ck$-network), Comment. Math. Univ. Carolinae 61 (2), 257–263, 2020.
- [22] L.Q. Tuyen and O.V. Tuyen, A note on the hyperspace of finite subsets, Fasciculi Math. 65, 67–73, 2021.
- [23] N.V. Velichko, Symmetrizable spaces, Math. Notes 12, 784–786, 1972.
- [24] W.A. Wilson, On semi-metric spaces, Amer. J. Math. 53, 361–373, 1931.
- [25] P.F. Yan, On strong sequence-covering compact mappings, Northeast. Math. J. 14 (3), 341–344, 1998.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | August 15, 2023 |
| Publication Date | April 23, 2024 |
| Published in Issue | Year 2024 Volume: 53 Issue: 2 |