A counterexample to the Sura-Bura Theorem for non separated compact spaces

Question

Is there a $T_1$ compact space such that the class of quasi-components differs from the class of connected components? By the Sura-Bura theorem, this is not possible for $T_2$ compact spaces.

Answer 1

Just take a convergent sequence with two limits, let's say $\{0, 0'\} ∪ \{1/n: n ∈ \mathbb{N}\}$. All the points $1/n$ are isolated. The neighborhood base at $0$ is $\{0\} ∪ \{1/n: n > N\}$, $N ∈ \mathbb{N}$, and similarly at $0'$: $\{0'\} ∪ \{1/n: n > N\}$, $N ∈ \mathbb{N}$.

All the singletons $\{1/n\}$ are clopen, and the two zeros can't be separated by disjoint neighborhoods, so the quasi-components are $\{0, 0'\}$, $\{1/n\}$, $n ∈ \mathbb{N}$. At the same time the space $\{0, 0'\}$ is discrete, so it is not a connected component of our space.

Note that you may obtain this space as a quotient of the sum of two convergent sequences, and also as a quotient of a Hausdorff but not compact counterexample to the Sura-Bura theorem – of the space where the points $1/n$ are replaced by vertical segments of length $1$ at $1/n$.