Exercise 7(a), Section 31 of Munkres’ Topology 
Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (a) Show that if $X$ is Hausdorff, then so is $Y$.

My attempt:
Approach(1): Let $x,y \in Y$ with $x\neq y$. So $p^{-1}(\{x\}),p^{-1}(\{y\})$ is compact and $p^{-1}(\{x\}) \cap p^{-1}(\{y\})= p^{-1}(\{x\} \cap \{y\})$$= p^{-1}(\emptyset)=\emptyset$. By exercise 5 section 26, $\exists U,V\in \mathcal{T}_X$ such that $p^{-1}(\{x\}) \subseteq U$, $p^{-1}(\{y\})\subseteq V$ and $U\cap V=\emptyset$. By generalize hint/chapter 3 theorem 11.2 of Dugundji topology, $\exists R \in \mathcal{N}_x$ such that $p^{-1}(R)\subseteq U$ and $\exists S\in \mathcal{N}_y$ such that $p^{-1}(S)\subseteq V$. Since $U\cap V=\emptyset$, we have $p^{-1}(R) \cap p^{-1}(S)=\emptyset =p^{-1}(R\cap S)$. Since $p$ is surjective, $R\cap S=\emptyset$. Hence $\exists R,S\in \mathcal{T}_Y$ such that $x\in R$, $y\in S$ and $R\cap S=\emptyset$. Is this proof correct?
Approach(2): It’s easy to check, normal $\Rightarrow$ regular $ \Rightarrow$ $T_2$ $\Rightarrow$ $T_1$. By Exercise 6, Section 31 of Munkres’ Topology, $Y$ is $T_2$. Is this proof correct?
Note: In approach(1) we used $p^{-1}(y)$ is compact, $p$ is closed map and $p$ is surjective conditions. In approach(2) we used $p$ is continuous, closed and surjective conditions.
 A: Edit: I am expanding this instead of commenting since stack exchange doesn’t seem to like so many comments:
Approach 2 does not work. In order to use your cited theorem, you would need the following
1 $X$ is normal.
2 a continuous, closed, surjective function from $X$ to $Y$
You have the following:
1 $X$ is Hausdorff.
2 a continuous, closed, surjective function from $X$ to $Y$ that has this other strong property.
So, to use the first theorem, you would need to show that $X$ in your new theorem is regular. That is not the case. The original theorem is weaker then if you replace normal with Hausdorff. Meaning it applies to much fewer situations, specifically it does not apply to situations with $X$ is not normal as we have.
Admittedly, I would save us both some headache if I found an example a a Hausdorff space that is not regular, and a closed continuous surjective map that is not perfect such that $Y$ is not Hausdorff. Unfortunately that is probably a pretty nasty example and is beyond me at the moment.
For approach 1, assuming you applied each theorem correctly this looks right to me. My main question is that the “generalize hint” talks about maps between compact spaces, do you know why it works for Hausdorff spaces? Assuming the other theorems are used correctly this looks fine to me.
