A Hausdorff space is called extremally disconnected or extreme, if for every open set $U$ the closure $\overline U$ is open, too. The question, whether there are extremally disconnected topological groups which are not discrete seems difficult, see


Reznichenko, Evgenii; Sipacheva, Ol'ga Discrete subsets in topological groups and countable extremally disconnected groups. Proc. Amer. Math. Soc. 149 (2021), no. 6, 2655–2668.


But what about locally compact groups? So here is my question:


* Is every extremally disconnected locally compact group discrete?


Or at least, can the question be answered for abelian groups?



Are there extremally disconnected groups?
Yes: every locally compact ED group is discrete.


Indeed, for topological groups, ED passes to quotients and open subgroups.


Let by contradiction $G$ be a non-discrete ED locally compact group. Passing to an open subgroup, one can suppose that $G$ is connected-by-compact. Hence it is compact-by-Lie. The Lie quotient being ED, it is discrete. Hence, passing to an open subgroup, one can suppose that $G$ is compact. If $G$ is not totally disconnected, by Weyl it has a positive-dimensional Lie quotient, again a contradiction. So $G$ is profinite. Hence it has a metrizable infinite quotient (just pick a decreasing sequence of closed normal subgroups of finite index). This is a Cantor space, not ED, contradiction.


