1) I am curious if the condition that $X$ is a connected space is necessary for the above theorem for Bockstein Spectral Sequence? The proof is from McCleary's User Guide to Spectral Sequence, and I don't really see where connectedness is used.
2) What does natural with respect to spaces and continuous mappings mean? I have a very vague idea that it is related to natural transformations, that means the continuous mappings "respect the structure" of the $B_*^r$. Is this important at all, and what does it mean actually?
Thank you for your help.