Fixed points of homeomorphisms and basically disconnected groups
Keywords:
Fixed-point-free autohomeomorphism, basically disconnected group, $F$-group, $P$-spaceAbstract
It is proved that if $X$ is a Tychonoff space with $\operatorname{dim}_0 X<\infty, h: X \rightarrow X$ is a fixed-point-free homeomorphism, and there exists a coarser paracompact topology on $X$ with respect to which $h$ remains a homeomorphism, then the extension $\beta h$ of $h$ to $\beta X$ is fixed-point-free. Consequences for topological groups are derived. In particular, it is proved that any finite-dimensional $F$-group of countable pseudocharacter contains an open Boolean subgroup and that the existence of an $\omega$-representable basically disconnected group not being a $P$-space is equivalent to the existence of a nondiscrete Boolean basically disconnected group of countable pseudocharacter.
References
E. K. van Douwen, $\beta X$ and fixed-point free maps, Topol. Appl. 51 (1993), 191-195.
M. G. Charalambous, Dimension Theory: A Selection of Theorems and Counterexamples. Cham: Springer International, 2019.
R. Engelking, General Topology. Berlin: Heldermann, 1989.
L. Gillman and M. Jerison, Rings of Continuous Functions. New York: Springer, 1960.
A. Arhangel'skii and M. Tkachenko, Topological Groups and Related Structures. Amsterdam, Paris: Atlantis Press/World Sci. 2008.
S. M. Sirota, The product of topological groups and extremal disconnectedness. Math. USSR-Sb. 8 (1969), no. 2, 169-180.
E. Reznichenko and O. Sipacheva, Discrete subsets in topological groups and countable extremally disconnected groups. Proc. Amer. Math. Soc. 149 (2021), 2655-2668.
J. R. Isbell, Zero-dimensional spaces. Tohoku Math. J. (2) 7 (1955), nos. 1-2, 1-8.
W. W. Comfort, Neil Hindman, and S. Negrepontis, $F^{\prime}$-Spaces and their product with $P$-spaces. Pacif. J. Math. 28 (1969), no. 3, 489-502.
O. Sipacheva, Topological groups with strong disconnectedness properties. Topol. Appl., Available online 14 December 2022, 108379. doi 10.1016/j.topol.2022.108379
A. V. Arkhangel'skii, Classes of topological groups. Russian Math. Surveys 36 (1981), no. 3, 151-174.
G. I. Kats, An isomorphic mapping of topological groups into a direct product of groups satisfying the first axiom of countability. Uspekhi Mat. Nauk 8 (1953), no. 6, 107-113.
