If $S \subset X$ is a connected subspace, is the set of all limit points of $S$ connected as well?

Question

I tried proving this and couldn't so I think it might be false, but I haven't thought of a counterexample yet.

score 3 · Accepted Answer · answered Feb 01 '19 at 00:25

3

The closure of a connected set is connected. Note that, in a connected set, every point is an accumulation point, so every point in the closure is an accumulation point. Thus, the set of accumulation points is the closure, which is connected.

answered Feb 01 '19 at 00:25

Theo Bendit

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How is it that every point in a connected set is an accumulation point? – Matheus Andrade Feb 01 '19 at 00:30
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@MatheusAndrade If $S$ is a subspace and $x \in S$ is not an accumulation point, then it's isolated, which makes ${x}$ and $S \setminus {x}$ open relative to $S$, and thus $S$ is disconnected. – Theo Bendit Feb 01 '19 at 00:32
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I see now, that was actually pretty obvious... thanks! – Matheus Andrade Feb 01 '19 at 00:33

If $S \subset X$ is a connected subspace, is the set of all limit points of $S$ connected as well?

1 Answers1