Are quasicomponents connected in a compact non-Hausdorff space?

Question

Are quasicomponents connected in a compact space?

Background:

• Quasicomponents are connected in a compact Hausdorff space.
• A non-compact locally compact Hausdorff space may not have connected quasicomponents.
• It can be shown that a subset is an open connected component if and only if it is an open quasicomponent. (*)
• In a finitely generated space (aka Alexandrov discrete space), every intersection of open subsets is open. Hence every quasicomponent is open as an intersection of clopen subsets. By (*), the quasicomponents are connected.
• More generally, a space with open components (a sumconnected space) has connected quasicomponents by (*). This includes locally connected spaces, which include finitely generated spaces.

So this question is about whether the first result can be generalized to non-Hausdorff spaces.

Answer 1

The following counter-example is taken from A. Tarizadeh, Idempotents and connected components, Example 3.15 (but I'm pretty sure that it appeared elsewhere long before):

Define $Y = \{\frac{1}{n}: n \in \mathbb N\} \cup \{0\}$ as a subspace of $\mathbb R$ and define $X = Y \cup \{0^*\}$ by doubling $0$, i.e. X is the subspace of the line with two origins.

Obviously, $Y$ is compact, $T_1$. Moreover, every clopen set, which contains $0$, also contains $0^*$. As each $\{\frac{1}{n}: n \ge m\} \cup \{0, 0^* \}$ is clopen, the quasi-component of $0$ is $\{0, 0^* \}$. However, $\{0, 0^* \}$ is discrete, hence not connected.