On monotonic bijections on subgroups of R

• Autores: Raushan Z. Buzyakova
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 17, Nº. 2, 2016, págs. 83-91
• Idioma: inglés
• DOI: 10.4995/agt.2016.4116
• Enlaces
  • Texto completo
• Resumen

  • 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

• Referencias bibliográficas
  • P. Alexandroff and P. Urysohn, Uber nuldimensionale Punktmengen, Math Ann. 98 (1928), 89-106.
  • (http://dx.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.
  • J. van Mill, The Infinite-Dimensional Topology of Function Spaces, Elsevier, 2001