Fixed points of homeomorphisms and basically disconnected groups - Reznichenko, Evgenii; Sipacheva, Ol’ga (Topology Proceedings, 2024)
It is proved that if X is a Tychonoff space with dim_0 X<∞, h: X → 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 β h of h to β 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 ω-representable basically disconnected group not being a P-space is equivalent to the existence of a nondiscrete Boolean basically disconnected group of countable pseudocharacter.