Theorem 6.1.23 in Engelking's Topology book says that in a compact space $X$ each quasicomponent is connected. Quasicomponent means the intersection of all closedandopen 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 quasicomponent is connected?
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.

$\begingroup$ 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). $\endgroup$ – Rick Sternbach Dec 12 '18 at 9:39

$\begingroup$ And I'm not sure we should call it the Engelking theorem. $\endgroup$ – Rick Sternbach Dec 12 '18 at 9:40