Theorem 6.1.23 in Engelking's Topology book says that in a compact space $X$ each quasi-component is connected. Quasi-component means the intersection of all closed-and-open subsets of $X$ containing a given point. The proof uses normality of $X$, so $X$ must be Hausdorff. But what if $X$ is only $T_1$ compact? Is it still true that each quasi-component is connected?
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The answer is no. Let $X$ be any totally disconnected infinite compact Hausdorff space, e.g. various projective limits of finite discret spaces. Take a point $a$ in $X$ and consider the analogue of the line of double origins: take two copies of $X$ and glue all the pairs of identified points except $a$ and its copy $a'$. Let $Y$ be the resulting quotient space. Then it is easy to see that $Y$ is quasi compact and satisfies $T1$ while the quasi component of $a$ (or $a'$) is the two point set $\{a, a'\} $ equipped with discret topology.
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$\begingroup$ Personally, I think the real reason for this theorem to hold is that a compact Hausdorff space has a unique uniform structure which is compatible with its topology. See e.g., section 4 of chapter 2 of Bourbaki's general topology. And $T1$ uniform space is automatically Hausdorff (completely regular even). $\endgroup$ Dec 12, 2018 at 9:39
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$\begingroup$ And I'm not sure we should call it the Engelking theorem. $\endgroup$ Dec 12, 2018 at 9:40