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?



$T_1$ version of Engelking theorem?
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. 


