Induced mappings on the hyperspace of totally disconnected sets
-
José G. Anaya
Abstract
The symbol TD(X) denotes the hyperspace of all nonempty totally disconnected compact subsets of a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. For a mapping between Hausdorff spaces f : X → Y, define the induced mapping TD(f) : TD(X) → TD(Y) by TD(f)(A) = f(A) (the image of A under f). In the current paper, we study the relationships between the condition f belongs to a class of mappings between Hausdorff spaces 𝕄 and the condition TD(f) belongs to 𝕄.
The work of the second author was supported by Becas de Posgrado de CONACyT, Grant No. 798362.
Acknowledgement
We express our gratitude to the reviewer for their insightful comments and valuable suggestions, which significantly contributed to enhancing the overall quality and clarity of this paper.
Communicated by L’ubica Holá
References
[1] Arens, R.: Note on convergence in topology, Math. Mag. 23 (1950), 229–234.10.2307/3028991Search in Google Scholar
[2] Engelking, R.: General topology, rev. and compl. edition, Sigma Ser. Pure Math., Vol. 6, Berlin: Heldermann Verlag, 1989.Search in Google Scholar
[3] Escobedo, R.—Pellicer-Covarrubias, P.—Sánchez-Gutiérrez, V.: The hyperspace of totally disconnected sets, Glas. Mat. III. Ser. 55(1)(2020), 113–128.10.3336/gm.55.1.10Search in Google Scholar
[4] Franklin, S. P.: Spaces in which sequences suffice, Fund. Math. 57 (1965), 107–115.10.4064/fm-57-1-107-115Search in Google Scholar
[5] García, Y.—Maya, D.: Induced mappings on symmetric products of Hausdorff spaces, Math. Slovaca 72(5) (2022), 1287–1300.10.1515/ms-2022-0088Search in Google Scholar
[6] Illanes, A.: Open induced mappings, an example, Topology Proc. 53 (2019), 37–45.Search in Google Scholar
[7] Maya, D.—Pellicer-Covarrubias, P.—Pichardo-Mendoza, R.: Induced mappings on the hyperspace of convergent sequences, Topology Appl. 229 (2017), 85–105.10.1016/j.topol.2017.07.008Search in Google Scholar
[8] Michael, E.: Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182.10.1090/S0002-9947-1951-0042109-4Search in Google Scholar
[9] Van Douwen, E. K.: The product of a Fréchet space and a metrizable space, Topology Appl. 47(3) (1992), 163–164.10.1016/0166-8641(92)90026-VSearch in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- 10.1515/ms-2024-frontmatter4
- Intervals of posets of a zero-divisor graph
- Coalgebraic methods for Ramsey degrees of unary algebras
- On nonexistence of D(n)-quadruples
- Rees short exact sequences and preenvelopes
- Generalized discrete Grüss and related results with applications
- Radius problem associated with certain ratios and linear combinations of analytic functions
- Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions
- Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
- On a solvable four-dimensional system of difference equations
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- The Maxwell-Boltzmann-Exponential distribution with regression model
- New results for the Marshall-Olkin family of distributions
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