A note on the homeomorphism group of the rational numbers
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- by Wayne R. Park
- Proc. Amer. Math. Soc. 42 (1974), 625-626
- DOI: https://doi.org/10.1090/S0002-9939-1974-0341368-9
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Abstract:
Let Q be the rational numbers with the usual topology, $H(Q)$ the group of homeomorphisms of Q, ${\gamma _c}$ the convergence structure of continuous convergence, and $\sigma$ the coarsest admissible convergence structure which makes $H(Q)$ a convergence group. A counterexample is constructed to show that if $\kappa$ is a convergence structure on $H(Q)$ such that ${\gamma _c} \leqq \kappa \leqq \sigma$, then $\kappa$ is never principal, hence never topological.References
- Richard Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610. MR 19916, DOI 10.2307/2371787
- C. H. Cook and H. R. Fischer, On equicontinuity and continuous convergence, Math. Ann. 159 (1965), 94–104. MR 179752, DOI 10.1007/BF01360283
- H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269–303 (German). MR 109339, DOI 10.1007/BF01360965
- Wayne R. Park, Convergence structures on homeomorphism groups, Math. Ann. 199 (1972), 45–54. MR 331354, DOI 10.1007/BF01419575
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 625-626
- MSC: Primary 54A20; Secondary 57E05
- DOI: https://doi.org/10.1090/S0002-9939-1974-0341368-9
- MathSciNet review: 0341368