A glance into the anatomy of monotonic maps

Authors

  • Raushan Buzyakova

DOI:

https://doi.org/10.4995/agt.2021.12291

Keywords:

monotonic map, ordered topological spaces, topologically equivalent maps

Abstract

Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps.

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References

R. Buzyakova, On monotonic fixed-point free bijections on subgroups of $R$, Applied General Topology 17, no. 2 (2016), 83-91. https://doi.org/10.4995/agt.2016.4116

R. Buzyakova and J. West, Three questions on special homeomorphisms on subgroups of R and R∞, Questions and Answers in General Topology 36, no. 1 (2018), 1-8.

R. Engelking, General Topology, PWN, Warszawa, 1977.

H. Bennet and D. Lutzer, Linearly Ordered and Generalized Ordered Spaces, Encyclopedia of General Topology, Elsevier, 2004. https://doi.org/10.1016/B978-044450355-8/50087-8

D. Lutzer, Ordered Topological Spaces, Surveys in General Topology, G. M. Reed., Academic Press, New York (1980), 247-296. https://doi.org/10.1016/B978-0-12-584960-9.50014-6

J. van Mill, The infinite-dimensional topology of function spaces, Elsevier, Amsterdam, 2001.

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Published

2021-04-01

How to Cite

[1]
R. Buzyakova, “A glance into the anatomy of monotonic maps”, Appl. Gen. Topol., vol. 22, no. 1, pp. 1–10, Apr. 2021.

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Section

Articles