# $T_1$ version of Engelking theorem?

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?

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.
• 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). – Rick Sternbach Dec 12 '18 at 9:39