The Frame_bundle article of Wikipedia mentions that the topology of frame bundle (as a fiber bundle it should have a topology) is constructed "reversely" to make the "local trivialization candidates" homeomorphisms.
Suppose $e:E\to X$ is a vector bundle, $\pi:F(E)\to X$ is the "frame bundle candidate" (since this is in the process to define frame bundle), $\psi_i:\pi^{-1}(U_i)\to U_i\times GL(k,R)$ is the "local trivilization candidates".
Then topologies are defined on each $\pi^{-1}(U_i)$ to make them homeomorphic to $U_i\times GL(k,R)$. The problem is if $\pi^{-1}(U_i)$ and $\pi^{-1}(U_j)$ have different open sets in $\pi^{-1}(U_i)\cap\pi^{-1}(U_j)$ then how can a topology be defined on $F(E)$ that is compatible with both? Note that if any open sets are added to or removed from the topology on $\pi^{-1}(U_i)$, then $\psi_i:\pi^{-1}(U_i)\to U_i\times GL(k,R)$ won't be a homeomorphisms anymore. This seems to be what the article does, it define the topology on $F(E)$ with the final topology, i.e. removing "incompatible open sets" from the topologies for $\pi^{-1}(U_i)$ which IMO make $\psi_i$ not homeomorphic anymore.