In Hatcher's book, Vector bundles and K-theory. He states the following version of Leray-Hirsch's theorem:
Let $p:E \longrightarrow B$ be a fiber bundle with $E$ and $B$ compact Hausdorff and with fiber $F$ such that $K^\ast(F)$ is free. Suppose there exists class $c_1, \cdots, c_n \in K^\ast(E)$ that restrict to a basis of $K^\ast (F)$ in each fiber $F$. If either (a) $B$ is a finite cell complex or (b)$F$ is a finite cell complex having all cells of even dimension, then $K^\ast(E)$, as a module over $K^\ast (B)$, is free with basis {$c_1,\cdots,c_n$}.
Can we drop those finiteness for $B$ and $F$ assumptions?
In Spanier's book, Algebraic topology, a similar theorem is stated for singular homology and cohomology without the finiteness assumption. From my understanding, Spanier first prove the theorem for homology under some finiteness assumption so he can do induction by long exact sequences like what Hatcher did. Then he dropped the finiteness assumption by the fact that homology is compatible with colimits and he concluded the result for cohomology by using the universal coefficient theorem ($H^\ast(Y,G)\otimes M^\ast \cong H^\ast(Hom(\Delta(Y)) \otimes M, G)$) and the homology case.
The problem is, from what I heard, $K$-theory is a cohomology theory so I'm not sure if it's compatible with limits. (I know singular cohomology isn't.) And I don't know if it has a universal coefficient theorem like theorem relates it with some compactly supported theorem.
So please tell me if you know the answer or some reference. Thank you.