Discrete subsets in topological groups and countable extremally disconnected groups

Reznichenko, Evgenii; Sipacheva, Ol’ga

doi:10.1090/proc/13992

Discrete subsets in topological groups and countable extremally disconnected groups
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by Evgenii Reznichenko and Ol’ga Sipacheva PDF

Proc. Amer. Math. Soc. 149 (2021), 2655-2668 Request permission

Abstract:

In 1967 Arhangel’skii posed the problem of the existence in ZFC of a nondiscrete extremally disconnected topological group. The general case is still open, but we solve Arhangel’skii’s problem for the class of countable groups. Namely, we prove that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter; hence, such a group cannot be constructed in ZFC. We also prove that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one limit point, which gives a negative answer to Protasov’s question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed.

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