Consider a fibre bundle $E\rightarrow B$. I was wondering when we can say that the pair $(E,B)$ is a good pair? By a good pair $(A,X)$ I mean two spaces $X\subset A$ such that $X$ is closed and is a deformation retract of some neighbourhood of it in $A$.
In the smooth category, we can assume all spaces to be CW-complexes and hence if we can show that $B$ is a subcomplex of $E$, then we have that $(E,B)$ is a CW-pair and in particular a good pair. I feel like this is the case, but i'm not entirely sure (my knowledge of CW-complexes is not very good, but the answer to the question Does the total space of a fibre bundle have the homotopy type of a CW complex if the base and the fibers have? does make it feel plausible).
However, I would also be interested in the more general case where we work in the category of topological spaces and have no access to smooth structures and hence cannot resort to CW complexes.
My interest comes from the relation between reduced (co)homology of the quotient space $E/B$ and the relative (co)homology of the pair $(E,B)$.