Let $X$ be a topological space, and $\sim$ be an equivalence relation on $X$. Let $\pi:X\rightarrow X/\sim$ be the natural surjective set map. Put quotient topology on $X/\sim$ w.r.t. the topology on $X$ and the surjection $\pi$.
Let $\Delta$ be the diagonal of $(X/\sim)\times (X/\sim)$.
Now,
(Topology on $X/\sim$ is Hausdorff) $\Longleftrightarrow$ ($\Delta$ is closed in $(X/\sim)\times (X/\sim)$) $\Longleftrightarrow$ ($(\pi\times\pi)^{-1}(\Delta)$ is closed in $X\times X$).
Q.1 Is the second equivalence correct (due to quotient topology)?
Next, $(\pi\times\pi)^{-1}(\Delta)=\{(x,y):x\sim y\}$. Call relation $\sim$ on $X$ closed if $\{(x,y):x\sim y\}$ is closed subspace of $X\times X$. So, the above equivalence says: $$ (\mbox{Topology on $X/\sim$ is Hausdorff}) \Longleftrightarrow (\sim \mbox{ is closed relation on $X$}) $$ Question 2: Is this last assertion correct?
In the book Homology theory by Vick, the author proves:
If $\sim$ is closed relation on compact Hausdorff space, then quotient topology on $X/\sim$ is Hausdorff
I am confused why compactness is needed, since in above arguments, it does not appear. Why can't we say from above equivalence that if $\sim$ is closed relation, then quotient topology on $X/\sim$ is Hausdorff? I confused to understand my fault in proof-or-statement understanding.
