Compact-fibre bundle totals preserve paracompactness, and CW type under CW-type hypotheses

$S$ is Hausdorff. Let $s\mathrm{\ne }{s}^{\mathrm{\prime }}$ be points of $S$. If $p(s)\mathrm{\ne }p(s^{\mathrm{\prime }})$, choose disjoint open $U,V\subseteq B$ containing them, possible because $B$ is Hausdorff; then $p^{-1}(U)$ and $p^{-1}(V)$ are disjoint open subsets of $S$ containing $s$ and $s^{\mathrm{\prime }}$. If $p(s)=p(s^{\mathrm{\prime }})=b$, choose a chart domain $U_i$ containing $b$ and apply the homeomorphism ${\theta }_i$: the images lie in $U_i\times F$, which is Hausdorff by [F4] because $U_i\subseteq B$ and $F$ are Hausdorff; separating them there and applying ${\theta }_i^{-1}$ separates $s$ and $s^{\mathrm{\prime }}$ in $S$.

$S$ is paracompact. Let $\mathcal{W}$ be an open cover of $S$. For $b\in B$ the fiber $S_b\cong F$ is compact, so $\mathcal{W}$ has a finite subfamily covering $S_b$; fix such a finite list for every $b$ (this is the only choice made, and it is a choice of one set for every element of $B$). In a chart ${\theta }_i$ about $b$ the union of the listed open sets contains $\{b\}\times F$ in $U_i\times F$; the tube lemma [F2] applied with compact factor $F$ gives an open $V_b\ni b$ with $V_b\times F$ contained in that union, that is, $p^{-1}(V_b)$ is covered by the finitely many listed members of $\mathcal{W}$. The family $\{V_b\}$ is an open cover of the paracompact space $B$, so by [F3] it has a locally finite open refinement $\{V_{\beta }{\}}_{\beta \in J}$. Choose for each $\beta$ an index $b(\beta )$ with $V_{\beta }\subseteq V_{b(\beta )}$; then the family consisting of the open sets $W\cap p^{-1}(V_{\beta })$, for all $\beta$ and all $W$ in the finite list attached to $b(\beta )$, covers $S$ and refines $\mathcal{W}$, since $p^{-1}(V_{\beta })\subseteq p^{-1}(V_{b(\beta )})$ is covered by that finite list. It is locally finite: given $s\in S$ with $p(s)=b_0$, local finiteness of $\{V_{\beta }\}$ at $b_0$ supplies an open $N\ni b_0$ meeting only finitely many $V_{\beta }$, and then the neighborhood $p^{-1}(N)$ of $s$ meets $p^{-1}(V_{\beta })$ only for those finitely many $\beta$. Hence every open cover of $S$ has a locally finite open refinement, so $S$ is paracompact by [F3].

The compact-generation and CW-type clauses. Suppose first that $B$ is compactly generated. Since a paracompact Hausdorff space is regular, every point of a chart domain $U_i$ has an open neighborhood in $B$ whose closure lies in $U_i$; [F7] therefore makes $U_i$ compactly generated. The compact Hausdorff fiber $F$ is locally compact, so [F7] makes each ordinary product $U_i\times F$, and hence each open chart $p^{-1}(U_i)$, compactly generated. Compact generation is local on this open cover: if $A\subseteq S$ meets every compact subspace of $S$ in a closed set, then $A\cap p^{-1}(U_i)$ has the same property in the compactly generated chart and is closed there; the chart cover then makes $A$ closed in $S$. Thus $S$ is compactly generated. It is weak Hausdorff because it is Hausdorff by step 1.1, so it is CGWH. Independently, if $B$ and $F$ have the homotopy type of CW complexes, then [F6] makes $p:S\to B$ a Hurewicz fibration, and [F5] gives the homotopy type of a CW complex for $S$.

Consequences and boundary cases. The projective bundle of a numerable rank-$n$ bundle $E\to B$ with $n\ge 1$ is a numerable fiber bundle with fiber ${\mathbb{R}\mathbb{P}}^{n-1}$, and the flag bundle is a composite of such bundles, each with the compact CW complex ${\mathbb{R}\mathbb{P}}^j$ as fiber; if the initial base is paracompact Hausdorff CGWH of CW type, the three preceding steps apply at each stage and give the same conclusion for every intermediate total space. A fiber product over $B$ of finitely many such total spaces is a numerable bundle over $B$ with compact fiber, so the same clauses apply. If $F=\mathrm{\varnothing }$ then $S=\mathrm{\varnothing }$; if $F$ is a point then $S\cong B$ over $B$; if $B=\mathrm{\varnothing }$ then $S=\mathrm{\varnothing }$. In each case paracompactness, Hausdorffness, compact generation and the CW-type conclusion hold under their stated hypotheses because the empty space and $B$ itself have the corresponding properties. For projective fibers, the one-point case is ${\mathbb{R}\mathbb{P}}^0=\{\ast \}$ and the empty convention is ${\mathbb{R}\mathbb{P}}^{-1}=\mathrm{\varnothing }$; both are compact and have CW type.