Let \(p: X \to Y\) be a perfect map. (See Exercise 7 of §31.)
Show that if \(Y\) is paracompact, so is \(X\). [Hint: If \(\mathcal{A}\) is an open covering of \(X\), find a locally finite open covering of \(Y\) by sets \(B\) such that \(p^{-1}(B)\) can be covered by finitely many elements of \(\mathcal{A}\); then intersect \(p^{-1}(B)\) with these elements of \(\mathcal{A}\).]
Show that if \(X\) is a paracompact Hausdorff space, then so is \(Y\). [Hint: If \(\mathcal{B}\) is a locally finite closed covering of \(X\), then \(\ts{p(B) \mid B \in \mathcal{B}}\) is a locally finite closed covering of \(Y\).]
Solution
Let \(p:X\to Y\) be perfect.
Assume \(Y\) is paracompact, and let \(\mathcal U\) be an open cover of \(X\). For each \(y\in Y\), compactness of the fiber \(p^{-1}(y)\) gives finitely many members \[ U_{y,1},\ldots,U_{y,r(y)}\in\mathcal U \] covering that fiber. Put \(O_y=\bigcup_jU_{y,j}\). Since \(p\) is closed, \[ V_y=Y-p(X-O_y) \] is open, contains \(y\), and satisfies \(p^{-1}(V_y)\subset O_y\). Choose a locally finite open refinement \(\mathcal W\) of \(\{V_y\}\) covering \(Y\). For each \(W\in\mathcal W\), choose \(y(W)\) with \(W\subset V_{y(W)}\), and form \[ \mathcal R= \{p^{-1}(W)\cap U_{y(W),j}:W\in\mathcal W,\ 1\le j\le r(y(W))\}. \] This is an open refinement of \(\mathcal U\) and covers \(X\). It is locally finite: if a neighborhood \(N\) of \(p(x)\) meets only finitely many \(W\)’s, then \(p^{-1}(N)\) meets only the finitely many corresponding finite families in \(\mathcal R\). Hence \(X\) is paracompact.
Assume \(X\) is paracompact Hausdorff. A paracompact Hausdorff space is regular, and a perfect map carries regularity of \(X\) to \(Y\) by E-Y4MFU, so \(Y\) is regular. Let \(\mathcal U\) be an open cover of \(Y\). The pullback cover \(\{p^{-1}(U):U\in\mathcal U\}\) of \(X\) has a locally finite closed refinement \(\mathcal F\) covering \(X\) (the closed-refinement form of paracompactness).
For each \(F\in\mathcal F\), choose \(U(F)\in\mathcal U\) with \[ F\subset p^{-1}(U(F)). \] Since \(p\) is closed, every \(p(F)\) is closed and \(p(F)\subset U(F)\); the family \(\{p(F):F\in\mathcal F\}\) covers \(Y\).
It is locally finite. Fix \(y\in Y\). The compact fiber \(p^{-1}(y)\) has a neighborhood \(O\subset X\) meeting only finitely many members of \(\mathcal F\): cover the fiber by finitely many neighborhoods witnessing local finiteness and take their union. Then \[ N=Y-p(X-O) \] is an open neighborhood of \(y\) with \(p^{-1}(N)\subset O\). If \(N\cap p(F)\ne\varnothing\), then \(p^{-1}(N)\cap F\ne\varnothing\), so \(F\) is one of the finitely many members meeting \(O\). Thus \(\{p(F)\}\) is locally finite.
Therefore every open cover of the regular space \(Y\) has a locally finite closed refinement. By the closed-refinement characterization of paracompactness, \(Y\) is paracompact.