
Hausdorff and continuity
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 $\displaystyle X \times \{0,1\}$ define the partition composed of the pairs $\displaystyle \{(a,0), (a,1)\}$ for $\displaystyle a \in A$ and of singletons $\displaystyle \{(x,i)\}$ if $\displaystyle x \in X$ \ $\displaystyle A$ and $\displaystyle i \in \{0,1\}$.
Let R be the equivalence relation defined by the partition and let Y be the quotient space $\displaystyle [X \times \{0,1\}]$ / $\displaystyle R$ and let $\displaystyle p: X \times \{0,1\} \rightarrow Y$ be the quotient map.
(a) Prove that there exists a continuous map $\displaystyle f: Y \rightarrow X$ such that $\displaystyle f ( p(x,i)) = x$ for every $\displaystyle x \in X$ and $\displaystyle i \in \{0,1\}$.
(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).