Abstract
For a topological space X, let CL(X) be the set of all non-empty closed subset of X, and denote the set CL(X) with the Vietoris topology by \((CL(X), {\mathbb {V}})\). In this paper, we mainly discuss the hyperspace \((CL(X), {\mathbb {V}})\) when X is an infinite countable discrete space. As an application, we first prove that the hyperspace with the Vietoris topology on an infinite countable discrete space contains a closed copy of nth power of Sorgenfrey line for each \(n\in {\mathbb {N}}\). Then we investigate the tightness of the hyperspace \((CL(X), {\mathbb {V}})\) and prove that the tightness of \((CL(X), {\mathbb {V}})\) is equal to the set-tightness of X. Moreover, we extend some results about the generalized metric properties on the hyperspace \((CL(X), {\mathbb {V}})\). Finally, we give a characterization of X such that \((CL(X), {\mathbb {V}})\) is a \(\gamma \)-space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
A space X is called perfect if every closed subset of X is a \(G_\delta \)-set.
A space \((X, \tau )\) is called a \(\beta \)-space if there exists a function \(g: {\mathbb {N}}\times X\rightarrow \tau \) such that (i) for any \(x\in X\), we have \(g(n+1, x)\subset g(n, x)\) for any \(n\in {\mathbb {N}}\), (ii) for any \(x\in X\) and sequence \(\{x_{n}\}\) in X, if \(x\in g(n, x_{n})\) for each \(n\in {\mathbb {N}}\), then \(\{x_{n}\}\) has an accumulation point in X
A space X is said to have a base of countable order if there is a sequence \(\{{\mathcal {B}}_n\}\) of bases for X such that: Whenever \(x\in b_n\in {\mathcal {B}}_n\) and \(\{b_n\}\) is decreasing, then \(\{b_n: n\in \omega \}\) is a base at x. We use ‘BCO’ to abbreviate ‘base of countable order’.
A regular space X is called a p-space if there is a sequence \(\{{\mathscr {U}}_{n}\}\) of families of open sets in \(\beta X\) such that (1) each \({\mathscr {U}}_{n}\) covers X; (2) for each \(x\in X\), \(\bigcap _{n\in {\mathbb {N}}}\text{ st }(x, {\mathscr {U}}_{n})\subset X\). If we also have (3) for each \(x\in X\), \(\bigcap _{n\in {\mathbb {N}}}\text{ st }(x, {\mathscr {U}}_{n})=\bigcap _{n\in {\mathbb {N}}}\overline{\text{ st }(x, {\mathscr {U}}_{n})}\), then X is called a strict p-space.
A function \(d: X\times X\rightarrow {\mathbb {R}}^{+}\) is called symmetric on a set X if for each \(x, y\in X\), we have (1) \(d(x, y)=0\) if and only if \(x=y\) and (2) \(d(x, y)=d(y, x)\). A space \((X, \tau )\) is called symmetrizable if there exists a symmetric d on X such that the topology \(\tau \) given on X is generated by the symmetric d, that is, a subset \(U\in \tau \) if and only if for every \(x\in U\), there is \(\varepsilon >0\) such that \(B(x, \varepsilon )\subset U\).
A space \((X, \tau )\) is called quasi-developable if there exists a sequence \(\{{\mathscr {U}}_{n}\}\) of families consisting of open sets in X such that for each \(x\in U\in \tau \) there exists \(n\in {\mathbb {N}}\) such that \(x\in \text{ st }(x, {\mathscr {U}}_{n})\subset U\).
A space \((X, \tau )\) is called developable if there exists a sequence \(\{{\mathscr {U}}_{n}\}\) of families of open covers of X such that, for each \(x\in X\), \(\{\text{ st }(x, {\mathscr {U}}_{n})\}\) is an open neighborhood base of x in X. A regular developable space is called a Moore space;
A space X is said to have a \(G_{\delta }\)-diagonal if there is a sequence \(\{{\mathscr {U}}_{n}\}\) of open covers of X, such that, for each \(x\in X\), \(\{x\}=\bigcap _{n\in {\mathbb {N}}}\text{ st }(x, {\mathscr {U}}_{n})\).
A space \((X, \tau )\) is called a semi-stratifiable if, there exists a function \(F: {\mathbb {N}}\times \tau \rightarrow \tau ^{c}\) satisfying the following conditions: (1) \(U\in \tau \Rightarrow U=\bigcup _{n\in {\mathbb {N}}}F(n, U)\); (2) \(V\subset U\Rightarrow F(n, V)\subset F(n, U)\), where \(\tau ^{c}=\{F: F\subset X, X\setminus F\in \tau \}\).
A space \((X, \tau )\) is a \(\gamma \)-space if there exists a function \(g: \omega \times X \rightarrow \tau \) such that (i) \(\{g(n, x): n\in \omega \}\) is a base at x; (ii) for each \(n\in \omega \) and \(x\in X\), there exists \(m\in \omega \) such that \(y\in g(m, x)\) implies \(g(m, y)\subset g(n, x)\). By [12, Theorem 10.6(iii)], each \(\gamma \)-space is a \(D_0\)-space.
A family \({\mathscr {P}}\) in a space X is called a network for X if, for each \(x\in U\) with U open in X, there exists \(P\in {\mathscr {P}}\) such that \(x\in P\subset U\).
A space X is called a k-space if, for each \(A\subset X\), A is closed in X provided \(K\cap A\) is closed for each compact subset K of X.
A family \({\mathcal {B}}\) of open subsets of a space X is called an external base [2, Page 467] of a set \(Y\subset X\) if for every point \(y\in Y\) and every neighborhood U of y in X there exists \(V\in {\mathcal {B}}\) such that \(y\in V\subset U\).
References
Arhangel’skii, A.V., Bella, A.: Countable fan-tightness versus countable tightness. Comment. Math. Univ. Carol. 37(3), 565–576 (1996)
Arhangel’skiǐ, A.V., Tkachenko, M.: Topological Groups and Related Structures. Atlantis Press, Amsterdam (2008)
Aull, C.E.: Closed set countabality axioms. Indag. Math. 28, 311–316 (1966)
Costantini, C., Levi, S., Pelant, J.: Compactness and local compactness in hyperspaces. Topol. Appl. 123, 573–608 (2002)
Dai, M., Liu, C.: $D_1$-spaces and their metrization. Northeast. Math. J. 11(2), 215–220 (1995)
Engelking, R.: General Topology, completed Heldermann Verlag, Berlin (1989)
García-Ferreira, S., Ortiz-Castillo, Y. F.: The hyperspace of convergent sequences. Topol. Appl. 196, 795–804 (2015)
García-Ferreira, S., Rojas-Hernández, R.: Connectedness like properties of the hyperspace of convergent sequences. Topol. Appl. 230, 639–647 (2017)
García-Ferreira, S., Rojas-Hernández, R., Ortiz-Castillo, Y. F.: Categorical properties on the hyperspace of nontrivial convergent sequences. Topol. Proc. 52, 265–279 (2018)
García-Ferreira, S., Rojas-Hernández, R., Ortiz-Castillo, Y. F.: The Baire property on the hyperspace of nontrivial convergent sequences. Topol. Appl. 301, 107505 (2020)
Good, C., Macías., S.: Symmetric products of generalized metric spaces. Topol. Appl. 206, 93–114 (2016)
Gruenhage, G.: Generalized Metric Spaces. Handbook of Set-Theoretic Topology, pp. 423–501. North-Holland, Amsterdam (1984)
Gruenhage, G., Michael, E., Tanaka, Y.: Spaces determined by point-countable covers. Pac. J. Math. 113(2), 303–332 (1984)
Holá, L., Pelant, J.: Recent progress in hyperspace topologies. In: Hušek, M., van Mill, J. (eds.) Recent Progress in General Topology II, pp. 253–279. Elsevier, Amsterdam (2002)
Holá, L., Pelant, J., Zsilinszky, L.: Developable hyperspaces are metrizable. Appl. Gen. Topol. 4(2), 351–360 (2003)
Holá, L., Levi, S.: Decomposition properties of hyperspace topologies. Set-Valued Anal. 5, 309–321 (1997)
Kubo, M.: A note on hyperspaces by compact sets. Mem. Osaka Kyoiku Univ. Ser. III 27, 81–85 (1978)
Lin, F., Shen, R., Liu, C.: Generalized metric properties on hyperspaces with the Vietoris topology. Rocky Mt. J. Math. 51(5), 1761–1779 (2021)
Lin, J., Lin, F., Liu, C.: Hyperspace of finite unions of convergent sequences. Stud. Sci. Math. Hung. 58(4), 433–456 (2021)
Maya, D., Pellicer-Covarrubias, P., Pichardo-Mendoza, R.: Cardinal functions of the hyperspace of convergent sequences. Math. Slovaca 68(2), 431–450 (2018)
Maya, D., Pellicer-Covarrubias, P., Pichardo-Mendoza, R.: General properties of the hyperspace of convergent sequences. Topol. Proc. 51(2), 143–168 (2018)
Michael, E.: Topologies on spaces of subsets. Trans. Am. Math. Soc. 71, 152–182 (1951)
Naimpally, S.A., Peters, J.F.: Hyperspace Topologies, Topology with Applications: Topological Spaces via Near and Far. World Scientific, Singapore (2013)
Ntantu, I.: Cardinal functions on hyperspaces and function spaces. Topol. Proc. 10, 357–375 (1985)
Peng, L., Sun, Y.: A study on symmetric products of generalized metric spaces. Topol. Appl. 231, 411–429 (2017)
Sabella, R.R.: Spaces in which compact sets have countable local base. Proc. Am. Math. Soc. 48, 499–504 (1975)
Tang, Z., Lin, S., Lin, F.: Symmetric products and closed finite-to-one mappings. Topol. Appl. 234, 26–45 (2018)
Acknowledgements
The authors wish to thank the referees for carefully reading preliminary version of this paper and providing many valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rosihan M. Ali.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The second author is supported by the Key Program of the Natural Science Foundation of Fujian Province (No. 2020J02043), the NSFC (No. 11571158), the lab of Granular Computing, the Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics.
Rights and permissions
About this article
Cite this article
Liu, C., Lin, F. A Note on Hyperspaces by Closed Sets with Vietoris Topology. Bull. Malays. Math. Sci. Soc. 45, 1955–1974 (2022). https://doi.org/10.1007/s40840-022-01349-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-022-01349-2
Keywords
- Hyperspace
- Countable set-tightness
- Compact metrizable
- \(\gamma \)-space
- Weakly first-countable
- \(D_{1}\)-space
- \(D_{0}\)-space