# Hausdorff and continuity

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 $i \in \{0,1\}$.