Mapping cylinder is Hausdorff 
If $f:X\to Y$ is a map between Hausdorff spaces, then $M_f$ and $C_f$ are Hausdorff. Here, $M_f$ and $C_f$ are mapping cyliner and mapping cone i.e. $M_f = X\times I\sqcup Y/(x,0)\sim f(x)$ for $x\in X$ and $C_f = M_f/(X\times\{1\})$

I first tried to prove $M_f$ is $T_2$. Before, as $X$ and $Y$ are $T_2$, $X\times I\sqcup Y$ is $T_2$. Now let $[x],[y]\in M_f$ such that $[x]\neq[y]$. Then $x\neq y$ so we can find open nbds $U,V$ of $x,y$ in $X\times I\sqcup Y$ respectively such that $U\cap V = \emptyset$. I have no idea how to get further. Could you give any hint?
 A: Hint: distinguish cases.
The classes in $M_f$ can be singletons, for most points this is the case, i.e. all $[(x,t)]$ where $t \neq 0$ are their own class.
If we have the class of $(x,0)$: this is a non-trivial class as $(x,0) \sim f(x)$ by definition, and the class becomes $(f^{-1}[\{f(x)\}] \times \{0\}) \cup \{f(x)\}$: it will also
contain all $(x',0)$ where $f(x')=f(x)$, as these are forced to be in the same class via transivity and the connecting $f(x)$.
How would you separate the points $\{(x,t)\}$ and $(f^{-1}[\{f(p)\}] \times \{0\}) \cup \{f(p)\}$ in $M_f$? The former has a saturared neighbourhood $X \times (a,1)$ in $X \times I \sqcup Y$ for some $0 <a < t$, while for the latter we can use the saturated neighbourhood $Y \sqcup X \times [0,a)$ in that same space. The quotient images of these disjoint saturated open sets are still disjoint (general fact) and as required.
A little more thougth is required to separate $(f^{-1}[\{f(x)\}] \times \{0\}) \cup \{f(x)\}$ and $(f^{-1}[\{f(x')\}] \times \{0\}) \cup \{f(x')\}$ for distinct $x,x'$ in $X$ with $f(x) \neq f(x')$; here you must use the Hausdorffness of $X$ and $Y$ really.
The class of some $y \in Y$ is either of the same form as the previous case, namely $\{y\} \cup (f^{-1}[\{y\}])\times \{0\}$ if $y\in f[X]$, or else just $\{y\}$.
I'll leave you to puzzle out the details, but you do need to go in such case by case detail and realise what the classes actually are as subsets of the sum.
