On monotonic bijections on subgroups of R

Authors

  • Raushan Buzyakova

DOI:

https://doi.org/10.4995/agt.2016.4116

Keywords:

ordered group, topological group, homeomorphism, shift, monotonic function, fixed point, periodic point

Abstract

We show that for any continuous monotonic  bijection $f$ on a $\sigma$-compact subgroup  $G\subset \mathbb R$ there exists a binary operation $+_f$ such that $\langle G, +_f\rangle$  is a topological group topologically isomorphic to $\langle G, +\rangle$ and $f$ is a shift with respect to $+_f$. We then show that monotonicity cannot be replaced by a periodic-point free continuous bijections. We explore a few routes leading to generalizations and counterexamples

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References

P. Alexandroff and P. Urysohn, Uber nuldimensionale Punktmengen, Math Ann. 98 (1928), 89-106. https://doi.org/10.1007/BF01451582

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

C. Nicolas, private communication, 2009

W. Sierpinski, Sur une propriete topologique des ensembles denombrablesdense en soi, Fund. Math. 1 (1920), 11-16. https://doi.org/10.4064/fm-1-1-11-16

J. van Mill, The Infinite-Dimensional Topology of Function Spaces, Elsevier, 2001

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Published

2016-10-03

How to Cite

[1]
R. Buzyakova, “On monotonic bijections on subgroups of R”, Appl. Gen. Topol., vol. 17, no. 2, pp. 83–91, Oct. 2016.

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Section

Articles