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 the singletons $\{(x,i)\}$ if $x\in X\setminus A$ and $i \in \{0,1\}$.
Let $R$ be the equivalence relation defined by this partition, let $Y$ be the quotient space $[X \times \{0,1\}]/R$ and let $p:X \times \{0,1\} \to Y$ be the quotient map.
(a) Prove that there exists a continuous map $f:Y \to X$ such that $f \circ p(x,i)=x$ for every $x\in X$ and $i \in \{0,1\}$.
(b) Prove that Y is Hausdorff if and only if X is Hausdorff and A a closed subset of X.
So far I have done part (a), but am struggling with part (b). I am thinking I will have to use the fact that if a quotient space $Z/R$ is Hausdorff then the graph of the equivalence relation in $Z \times Z$ is closed. However, I am not positive. If anyone could even just help me get started somewhere on this question I would be grateful, thank you.
Best Answer
Hints: It actually shouldn't be too difficult to just go through the proof by hand without relying on other theorems.