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The standard example of a non-Hausdorff space which is $T_1$ is either a cofinite space over an infinite base set or a cocountable space over something uncountable. However, those topologies are already hyperconnected (we cannot cover $X$ by two distinct proper closed subsets), which implies that not only do there exist two points which cannot be separated by disjoint open subsets, but in fact no two points can be separated in that way (hint: reformulate the separability statement in terms of closed sets).

The obvious „fix“ to give a space that is not hyperconnected would be to take the disjoint union with any other second space, which of course retains the non-Hausdorffness.

But are there any connected examples?

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What about the line with two origins? Every point, including the two origins, is closed, but the two origins can't be separated by disjoint open sets. It is the union of two proper nonempty closed sets, but not two disjoint nonempty closed sets, so it is connected but not hyperconnected.

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  • $\begingroup$ That is not $T_0$ (and, in particular, not $T_1$) because $p$ and $q$ are topologically indistinguishable, or am I missing something $\endgroup$ – Lukas Juhrich Oct 28 '19 at 20:42
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    $\begingroup$ @Lukas It's $T_0$. The two origins can be separated but not by disjoint open sets. The open sets can contain one origin or the other, not necessarily both. $\endgroup$ – Matt Samuel Oct 28 '19 at 20:43
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There is another "obvious fix" that does preserve connectedness--take a wedge sum with a second space. If $X$ is any connected non-Hausdorff $T_1$ pointed space and $Y$ is any connected $T_1$ pointed space with more than one point, then $X\vee Y$ is connected, $T_1$, and non-Hausdorff, but not hyperconnected since $X$ and $Y$ are both closed in it.

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