Real projective bundle and tautological line

The projectivized transitions are well defined and obey the cocycle law. For $x\in U_i\cap U_j$ the linear isomorphism $g_{ji}(x)$ carries lines to lines and depends continuously on $x$, so the joint map $(x,\mathrm{\ell })\mapsto [g_{ji}(x)]\mathrm{\ell }$ is continuous. Indeed on a projective coordinate chart choose the representative with a specified coordinate equal to one, apply the continuous matrix, and take its nonzero-vector projective quotient. The identities of [F1] give $[g_{ii}(x)]=\mathrm{id}$ and $[g_{ki}(x)]=[g_{kj}(x)][g_{ji}(x)]$ for $x\in U_i\cap U_j\cap U_k$, because projectivization is functorial for composition of linear isomorphisms. Reading the displayed relation on the overlaps, the cocycle law makes $\sim$ reflexive, symmetric and transitive: it is the same calculation as in [F2] with $[g_{ji}]$ in place of $g_{ji}$. Therefore the quotient $P(E)$ exists, and the induced projection $p:P(E)\to B$ is continuous by [F4], since its composite with the quotient map is the first-coordinate projection on each summand.

The quotient is a numerable fiber bundle with fiber ${\mathbb{R}\mathbb{P}}^{n-1}$. Let $Q:{\coprod }_i{U}_i\times {\mathbb{R}\mathbb{P}}^{n-1}\to P(E)$ be the quotient map. It is open: if $O$ is open in the disjoint union, then on the $j$-th summand the saturation $Q^{-1}Q(O)$ is the union, over $i$, of the images of $O\cap ((U_i\cap U_j)\times {\mathbb{R}\mathbb{P}}^{n-1})$ under the overlap homeomorphisms $(x,\mathrm{\ell })\mapsto (x,[g_{ji}(x)]\mathrm{\ell })$, and is therefore open. Thus $Q(O)$ is open by the definition of the quotient topology. It follows that the restrictions of $Q$ to the $i$-th summands are open onto $p^{-1}(U_i)$. The maps ${\mathrm{\Phi }}_i:p^{-1}(U_i)⟶U_i\times {\mathbb{R}\mathbb{P}}^{n-1},{\mathrm{\Phi }}_i[x,\mathrm{\ell },k]=(x,[g_{ik}(x)]\mathrm{\ell })$ are well defined, continuous by [F4], and inverse over $U_i$ to the maps $(x,\mathrm{\ell })\mapsto [x,\mathrm{\ell },i]$; the latter maps are open by the preceding calculation. Hence they are homeomorphisms over $U_i$, so $P(E)$ is a locally trivial fiber bundle with fiber ${\mathbb{R}\mathbb{P}}^{n-1}$ in the sense of [F3]. The given numerating cover and partition of unity of $E$ serve unchanged, since the chart domains are the same $U_i$ and their supports are already subordinate; hence $P(E)$ is numerable.

The tautological quotient has the local bundle descriptions $U_i\times {\gamma }^{n-1}$: the same open-saturation argument as in step 2.1 applies to the overlap homeomorphisms $(x,\mathrm{\ell },v)\mapsto (x,[g_{ji}(x)]\mathrm{\ell },g_{ji}(x)v)$. Inside each such description refine the base by $D_{i,a}=\{(x,\mathrm{\ell }):v_a\mathrm{\ne }0\text{ for }0\mathrm{\ne }v\in \mathrm{\ell }\}$, for $1\le a\le n$. The unique vector $w_a(\mathrm{\ell })\in \mathrm{\ell }$ with coordinate $a$ equal to one is a continuous nonzero section. The map $(x,\mathrm{\ell },t)\mapsto (x,\mathrm{\ell },t{w}_a(\mathrm{\ell }))$ and its inverse, which reads the $a$-th coordinate of the vector, are continuous linear bundle charts. Thus the tautological quotient is a rank-one real bundle with the asserted fiber, not generally trivial over all of $p^{-1}(U_i)$.

If $B$ is paracompact Hausdorff CGWH of CW type and AC is assumed, [F5] applies to the numerable projective bundle of step 2.1: the fiber is the compact Hausdorff finite CW space ${\mathbb{R}\mathbb{P}}^{n-1}$. It follows directly that $P(E)$ is paracompact Hausdorff, CGWH, and of CW type, which is the asserted total-space consequence.

An explicit refined numeration needs no choice. In chart $i$ put $s_{i,a}(\mathrm{\ell })=v_a^2\mathrm{/}{\sum }_b{v}_b^2$, independent of the nonzero representative, and $d_{i,a}=\max (s_{i,a}-1\mathrm{/}(2n),0)$. Since some $s_{i,a}\ge 1\mathrm{/}n$, the sum $D_i={\sum }_a{d}_{i,a}$ is positive. Put ${\theta }_{i,a}=d_{i,a}\mathrm{/}{D}_i$, and define ${\psi }_{i,a}=({\rho }_i\circ p){\theta }_{i,a}$ on $p^{-1}(U_i)$, extended by zero elsewhere. This extension is continuous since points outside $U_i$ have a neighborhood disjoint from the closed support of ${\rho }_i$. The family is locally finite, because the base family is locally finite and there are only $n$ coordinates for each $i$. Its sum is one. Its closed support lies in $p^{-1}(\mathrm{supp}{\rho }_i)$ and in the locus $s_{i,a}\ge 1\mathrm{/}(2n)$, hence inside $D_{i,a}$. It is therefore support-subordinate to the actual line charts of step 3.1, proving numerability of ${\gamma }_E$.

For $n=1$, the projective fiber is a point, so the displayed charts identify $P(E)$ with $B$ and ${\gamma }_E$ with $E$. Rank zero uses only the declared empty-space convention, not the formulas involving $1\mathrm{/}(2n)$. An empty base gives empty quotients. Steps 1.1, 2.1, 3.1 and 4.1 use the supplied charts and partition and finite coordinate operations only; AC enters solely in step 3.2 through [F5].