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.
[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.
[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.
Tuyen, L. Q., Ong Van, T., & Kočinac, L. D. R. (2024). The Vietoris hyperspace \mathcal F(X) and certain generalized metric properties. Hacettepe Journal of Mathematics and Statistics, 53(2), 356-366. https://doi.org/10.15672/hujms.1203236
AMA
Tuyen LQ, Ong Van T, Kočinac LDR. The Vietoris hyperspace \mathcal F(X) and certain generalized metric properties. Hacettepe Journal of Mathematics and Statistics. April 2024;53(2):356-366. doi:10.15672/hujms.1203236
Chicago
Tuyen, Luong Quoc, Tuyen Ong Van, and Ljubiša D. R. Kočinac. “The Vietoris Hyperspace \mathcal F(X) and Certain Generalized Metric Properties”. Hacettepe Journal of Mathematics and Statistics 53, no. 2 (April 2024): 356-66. https://doi.org/10.15672/hujms.1203236.
EndNote
Tuyen LQ, Ong Van T, Kočinac LDR (April 1, 2024) The Vietoris hyperspace \mathcal F(X) and certain generalized metric properties. Hacettepe Journal of Mathematics and Statistics 53 2 356–366.
IEEE
L. Q. Tuyen, T. Ong Van, and L. D. R. Kočinac, “The Vietoris hyperspace \mathcal F(X) and certain generalized metric properties”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, pp. 356–366, 2024, doi: 10.15672/hujms.1203236.
ISNAD
Tuyen, Luong Quoc et al. “The Vietoris Hyperspace \mathcal F(X) and Certain Generalized Metric Properties”. Hacettepe Journal of Mathematics and Statistics 53/2 (April 2024), 356-366. https://doi.org/10.15672/hujms.1203236.
JAMA
Tuyen LQ, Ong Van T, Kočinac LDR. The Vietoris hyperspace \mathcal F(X) and certain generalized metric properties. Hacettepe Journal of Mathematics and Statistics. 2024;53:356–366.
MLA
Tuyen, Luong Quoc et al. “The Vietoris Hyperspace \mathcal F(X) and Certain Generalized Metric Properties”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, 2024, pp. 356-6, doi:10.15672/hujms.1203236.
Vancouver
Tuyen LQ, Ong Van T, Kočinac LDR. The Vietoris hyperspace \mathcal F(X) and certain generalized metric properties. Hacettepe Journal of Mathematics and Statistics. 2024;53(2):356-6.