I have often heard quite a few times that they people talk about, say, "the" twisted S^2-bundle $S^2$-bundle over S^2. $S^2$. My question is, what do they mean by a twisted bundle? I know that in the above example any S^2-bundle $S^2$-bundle over S^2 $S^2$ is either S^2 x $S^2 or \times S^2$ or a unique non-trivial S^2-bundle $S^2$-bundle over S^2. but $S^2$. But is that how it is defined? I mean is Is a twisted bundle just a non-trivial one? , or is it a specific bundle among the non-trivial ones?