*Based upon discussion at Math.SE*

Consider the property extremally disconnected, for which the closure of any open set remains open.

Frequently, this property is paired with the assumption of Hausdorff. This allows nice results like all extremally disconnected spaces are totally separated and disallows silliness like all hyperconnected spaces are extremally disconnected.

But I like silliness, so how much of the theory can we recover without Hausdorff? In this Math.SE post it was pointed out that while all Hausdorff extremally disconnected spaces are sequentially discrete (only trivial sequences converge), the cofinite topology on an infinite set is a $T_1$ extremally disconnected space that is not sequentially discrete.

On the other hand, the only example I know of a sequentially discrete, $T_1$, extremally disconnected space is US (limits of sequences are unique): the cocountable topology on an uncountable set. In fact, this is pretty immediate: all spaces where countable sets are closed are sequentially discrete, and all sequentially discrete spaces are US.

Might it be the case that the theorem that all Hausdorff extremally disconnected spaces are sequentially discrete can be improved to only assume US? If not, what counterexample can be constructed?

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