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

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 $1/n$.