If there are still some people following the set theory chatroom, perhaps this might be of interest for them: Cardinality of non-trivial compact (perfect or connected) Hausdorff spaces
Just in case the other room is deleted for inactivity, I'll copy dfeuer's question here:
I've had the vaguest sort of notion of how one might think about trying to construct a counter-example, but I'm stuck. The notion is that if DC does not hold, then there is an infinite sequence of non-empty sets such that there is no sequence of elements of those sets in that order. Would it be possible, somehow, to construct a topology on the image of that sequence?
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