Past exam of the options II for mathematics and philosophy course of the University of Oxford / 2013 / E4 / c / iii / Solution

If $A$ is disconnected, take a separation $A=A_1\cup A_2$. Each piece is also relatively closed, being the complement of the other. They are compact as closed subsets of the compact space $A$.

Part (b), using the ambient Hausdorff property, gives disjoint ambient open neighborhoods of $A_1$ and $A_2$. They meet $A$, cover it, and are disjoint everywhere, as required. Conversely, such ambient open sets give a separation on intersecting with $A$, by part (ii). Hence compactness and the Hausdorff property permit fully disjoint ambient neighborhoods.