= 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 \b[<compactness> and the Hausdorff property permit fully disjoint ambient neighborhoods].
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