I have been given a question that aims to demonstrate that the intersection of two compact subsets in a space that is not Hausdorff may not be compact and not closed. It takes you through step by step, but I can't do the first part:

Let X be a topological space and A a subset of X. On X \times \{0,1\} define the partition composed of the pairs \{(a,0), (a,1)\} for a \in A and of singletons \{(x,i)\} if x \in X \ A and i \in \{0,1\}.

Let R be the equivalence relation defined by the partition and let Y be the quotient space [X \times \{0,1\}] / R and let p: X \times \{0,1\} \rightarrow Y be the quotient map.

(a) Prove that there exists a continuous map f: Y \rightarrow X such that f ( p(x,i)) = x for every x \in X and i \in \{0,1\}.

(b) Prove that Y is Hausdorff iff X is Hausdorff and A is a closed subset of X. (For the direct implication I know that is the quotient space is Hausdorff the the graph of the equivalence relation is closed).