First, to add more context, let me write the theorem in question:
Cut wire theorem: If $(X, \tau)$ is a compact Hausdorff space and $A, B \subseteq X$ are two different closed sets such that every connected component of $X$ intersects at most one of them, then there exists $ U, V \in \tau$ disjoint, such that $A \subseteq U,\, B \subseteq V,\,\text{ and } U \cup V = X$
I'm trying to find some spaces where this doesn't hold. One of them Hausdorff but not compact and the other one compact but not Hausdorff. I would appreciate any help to find spaces like that. My only idea about this is that such space must be disconnected wich is not quite helpful. Thanks
A standard example for the first case is the subspace of the plane consisting of the lines $[0,1] \times \{ 1/n \}$, $n \in \mathbb N$, and the singletons $\{(0,0)\}$ and $\{(1,0)\}$. The two singletons are connected components but lie in the same quasi-component.
For a non-Hausdorff example, take the quotient of above formed by shrinking each of the lines to a point. (A sequence with two limit points.)