Paracompactness and Perfect Maps (6.41.8)

James Munkres Topology, chapter 6.41, exercise 8:

Let $p:X→Y$ be a closed surjective continuous map such that $p^{-1}\{y\}$ is compact for each $y∈Y$. If $X$ is Hausdorff, show $X$ is paracompact if and only if $Y$ is.

Proof: ($⇐$) Let $Y$ be paracompact, and let $\mathcal{B}=\{U_α\}$ be an open cover of $X$. For each $y∈Y$, let $\mathcal{B}_y⊆\mathcal{B}$ be a finite subcover of $p^{-1}\{y\}$, and obtain open $V_y⊆X$ such that $p^{-1}\{y\}⊆V_y⊆∪\mathcal{B}_y$ and $p(V_y)⊆Y$ is a neighborhood of $y$. Then $\{p(V_y)\}$ is an open cover for $Y$, so let $\mathcal{A}$ be a locally finite open refinement covering $Y$. For each $y∈Y$, let $y∈A_y∈\mathcal{A}$. We claim $\mathcal{C}=\{p^{-1}(A_y)∩B~|~y∈Y, B∈\mathcal{B}_y\}$ is a locally finite open refinement of $\mathcal{B}$ covering $X$. It is clear $\mathcal{C}$ is an open refinement, and given $x∈X$, we have $x∈p^{-1}(y)∩B⊆p^{-1}(A_y)∩B$ for some $y∈Y$ and $B∈\mathcal{B}_y$, therefore $\mathcal{C}$ covers $X$. As well, let $U$ be a neighborhood of $p(x)$ intersecting $A_y$ for only finitely many $y∈Y$. Then $p^{-1}(U)$ is a neighborhood of $x$ intersecting $p^{-1}(A_y)$ for only finitely many $y∈Y$, hence intersecting $p^{-1}(A_y)∩B$ for only finitely many pairs $(y,B)$ when $B∈\mathcal{B}_y$. Therefore $\mathcal{C}$ is locally finite and $X$ is paracompact.

($⇒$) Let $X$ be paracompact, and let $\mathcal{B}$ be an open cover of $Y$. Then let $\mathcal{A}$ be a locally finite open refinement of $\{p^{-1}(B)~|~B∈\mathcal{B}\}$ covering $X$. We claim $\mathcal{C}=\{p(A)~|~A∈\mathcal{A}\}$ is a locally finite refinement of $\mathcal{B}$ covering $Y$, so that since $Y$ is regular by normality of $X$, $Y$ will be paracompact by Lemma 41.3. The only nontrivial quality to check is local finiteness; given $y∈Y$, for each $x∈p^{-1}(y)$ let $U$ be a neighborhood of $x$ intersecting only finitely many elements of $\mathcal{C}$. Since $p^{-1}(y)$ is compact, there exists a neighborhood about it intersecting only finitely many elements of $\mathcal{C}$, and furthermore a saturated sub-neighborhood $V_y$ of this one. Being saturated, $V_y∩A=ø$ implies $p(V_y)∩p(A)=ø$ for all $A∈\mathcal{A}$, so that $p(V_y)$ is a neighborhood of $y$ intersecting only finitely many elements of $\mathcal{C}$.$~\square$