The fixed-locus Borel base is paracompact and of CW type

If $r=0$, $BT$ is a point and the CW and paracompact assertions reduce to the compact manifold $F$. For $r\ge 1$, [F1] gives $BT$ its ordinary product CW topology, and [F3] makes it paracompact Hausdorff. By [F2], choose a finite CW complex $K$ and a homotopy equivalence $h:F\to K$. Since $K$ is finite, its CW product with the CW complex $BT$ has the ordinary product topology; hence $BT\times K$ is a CW complex. The maps ${\mathrm{i}\mathrm{d}}_{BT}\times h$ and the product of a homotopy inverse with ${\mathrm{i}\mathrm{d}}_{BT}$ are homotopy inverses, since their coordinate homotopies are continuous in the product topology. Thus $BT\times F$ has CW type. If $F=\mathrm{\varnothing }$, take $K=\mathrm{\varnothing }$.

If $F=\mathrm{\varnothing }$, the product is empty and paracompact Hausdorff, so assume $F\mathrm{\ne }\mathrm{\varnothing }$. To prove paracompactness of $BT\times F$, let $\mathcal{U}$ be an open cover. For each $x\in BT$ and $y\in F$, choose a product neighborhood $O_{x,y}\times V_{x,y}$ of $(x,y)$ contained in some member $U_{x,y}\in \mathcal{U}$. Compactness of $F$ selects finitely many $y_1,\dots ,y_m$ whose $V_{x,y_j}$ cover $F$. Then $O_x={\bigcap }_{j=1}^m{O}_{x,y_j}$ is an open neighborhood of $x$, and $O_x\times F$ is covered by $U_{x,y_1},\dots ,U_{x,y_m}$. Paracompactness of $BT$ gives a locally finite open refinement $\{W_{\alpha }\}$ of $\{O_x\}$. For each $\alpha$, choose $x_{\alpha }$ with $W_{\alpha }\subseteq O_{x_{\alpha }}$; the finitely many sets $(W_{\alpha }\times F)\cap U_{x_{\alpha },y_j}$ cover $W_{\alpha }\times F$, are open, and refine $\mathcal{U}$. They are locally finite because a neighborhood $N\times F$ meets sets indexed by only those $\alpha$ for which $W_{\alpha }$ meets $N$, a finite set. Hence $BT\times F$ is paracompact. It is Hausdorff as a product of Hausdorff spaces. The point and rank-zero cases satisfy the same conclusions.