Continuous functions on primal topological spaces induced by group actions

Luis Fernando Mejías; Jorge Vielma; Elvis Aponte; Lourival Rodrigues De Lima; Luis Fernando Mejías; Jorge Vielma; Elvis Aponte; Lourival Rodrigues De Lima

doi:10.3934/math.2025037

AIMS Mathematics

2025, Volume 10, Issue 1: 793-808. doi: 10.3934/math.2025037

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Research article

Continuous functions on primal topological spaces induced by group actions

Departamento de Matemáticas, Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, ESPOL, Campus Gustavo Galindo, km. 30.5 vía Perimetral, Guayaquil, 090902, Ecuador

Received: 10 October 2024 Revised: 09 November 2024 Accepted: 14 November 2024 Published: 14 January 2025
MSC : Primary 54A10, 54C05; Secondary 54D05, 54D30

If $ G $ is a group acting on a set $ X $, then for any $ a\in G $, the restriction $ \phi_a:X\to X $ of the action to $ a $ induces a topology $ \tau_a $ for $ X $, called the primal topology induced by $ \phi_a $. First, we obtain a characterization of the normal subgroups in terms of the primal topologies. Later, we prove that some commutative relations among elements on the group $ G $ determine the continuity of maps among different primal spaces $ (X, \tau_{ \phi_x}) $. In particular, we prove the continuity of some maps when $ a, b, q\in G $ satisfy a quantum type relation, $ ba = qab $, as is in the quaternion and Heisenberg groups.
- group action,
- primal topology,
- continuous function
Citation: Luis Fernando Mejías, Jorge Vielma, Elvis Aponte, Lourival Rodrigues De Lima. Continuous functions on primal topological spaces induced by group actions[J]. AIMS Mathematics, 2025, 10(1): 793-808. doi: 10.3934/math.2025037

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Abstract

If $ G $ is a group acting on a set $ X $, then for any $ a\in G $, the restriction $ \phi_a:X\to X $ of the action to $ a $ induces a topology $ \tau_a $ for $ X $, called the primal topology induced by $ \phi_a $. First, we obtain a characterization of the normal subgroups in terms of the primal topologies. Later, we prove that some commutative relations among elements on the group $ G $ determine the continuity of maps among different primal spaces $ (X, \tau_{ \phi_x}) $. In particular, we prove the continuity of some maps when $ a, b, q\in G $ satisfy a quantum type relation, $ ba = qab $, as is in the quaternion and Heisenberg groups.

References

[1]	I. Dahane, S. Lazaar, T. Richmond, T. Turki, On resolvable primal spaces, Quaest. Math., 42 (2019), 15–35. https://doi.org/10.2989/16073606.2018.1437093
[2]	O. Echi, The categories of flows of set and top, Topol. Appl., 159 (2012), 2357–2366. https://doi.org/10.1016/j.topol.2011.11.059 doi: 10.1016/j.topol.2011.11.059
[3]	O. Echi, T. Turki, Spectral primal spaces, J. Algebra Appl., 18 (2019), 1–12. https://doi.org/10.1142/S0219498819500300
[4]	D. B. Ellis, R. Ellis, M. Nerurkar, The topological dynamics of semigroup actions, Trans. Amer. Math. Soc., 353 (2001), 1279–1320. https://doi.org/10.1090/S0002-9947-00-02704-5 doi: 10.1090/S0002-9947-00-02704-5
[5]	A. Haouati, S. Lazaar, Primal spaces and quasihomeomorphisms, Appl. Gen. Topol., 16 (2015), 109–118. https://doi.org/10.51768/dbr.v16i1.161201510 doi: 10.51768/dbr.v16i1.161201510
[6]	S. Lazaar, T. Richmond, H. Sabri, The autohomeomorphism group of connected homogeneous functionally Alexandroff spaces, Commun. Algebra, 7 (2019), 3818–3829. https://doi.org/10.1080/00927872.2019.1570240 doi: 10.1080/00927872.2019.1570240
[7]	S. Lazaar, H. Sabri, The topological group of autohomeomorphisms of homogeneous functionally Alexandroff spaces, Topology Proceedings, 55 (2020), 243–264.
[8]	L. Mejías, J. Vielma, A. Guale, E. Pineda, Primal topologies on finite-dimensional vector spaces induced by matrices, Int. J. Math. Math. Sci., 2023 (2023), 1–7. https://doi.org/10.1155/2023/9393234 doi: 10.1155/2023/9393234
[9]	B. Richmond, Principal topologies and transformation semigroups, Topol. Appl., 155 (2008), 1644–1649. https://doi.org/10.1016/j.topol.2008.04.007 doi: 10.1016/j.topol.2008.04.007
[10]	B. Richmond, Semigroups and their topologies arising from Green's left quasiorder, Appl. Gen. Topol., 9 (2008), 143–168. https://doi.org/10.4995/agt.2008.1795 doi: 10.4995/agt.2008.1795
[11]	S. Semmes, An introduction to Heisenberg groups in analysis and geometry, Notices Amer. Math. Soc., 50 (2003), 640–646.
[12]	F. A. Z. Shirazi, N. Golestani, Functional Alexandroff spaces, Hacet. J. Math. Stat., 40 (2011), 515–522.
[13]	J. Vielma, A. Guale, A topological approach to the Ulam-Kakutani-Collatz conjecture, Topol. Appl., 256 (2019), 1–6. https://doi.org/10.1016/j.topol.2019.01.012 doi: 10.1016/j.topol.2019.01.012
[14]	J. Vielma, A. Guale, F. Mejías, Paths in primal spaces and the Collatz conjecture, Quaest. Math., 43 (2020), 1–7. https://doi.org/10.2989/16073606.2020.1806939 doi: 10.2989/16073606.2020.1806939

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