I asked the same question in math stackexchange: <https://math.stackexchange.com/questions/2322883/how-can-i-endow-a-locally-product-cw-structure-on-a-vector-bundle-over-a-cw-co>
but it seems that it's harder than I thought, so I ask here:


I'm now learning characteristic classes, and I need a CW structure on the total space of a vector bundle $E\to B$ where $B$ is a CW complex such that the associated sphere bundle and rectriction over any subcomplex of $B$ are both subcomplexes (this is required in "Algebraic Topology from a Homotopical Viewpoint", page 364). I think this should be something like a locally product structure, but I couldn't figure out how to glue them together. I even doubt that this can be done.



How can I endow a "locally product" CW structure on a vector bundle over a CW complex?
The authors of this book are attempting to use CW structures to justify certain cohomology isomorphisms, but this seems to be the wrong approach since some of their claims about CW structures are just not true. For example, they say a vector bundle over a CW complex base space has a CW structure such that the complement of the zero section is a subcomplex, but this cannot be true since a subcomplex is always a closed subspace. It seems best not to talk about CW structures on vector bundles and instead prove the cohomology isomorphisms using standard tools such as excision for general topological spaces. 


