1. The problem statement, all variables and given/known data See attached picture. 2. Relevant equations 3. The attempt at a solution At the moment, I am dealing with part (a). What I am perplexed by is the ordering of the parts. If the subbasis in part (b) does indeed generate this coarsest topology, why wouldn't showing this be included in part (a)? What does the author intend I do for part (a), if not showing that the subbasis in part (a) generates this coarsest topology. To me it seems that, in order to properly solve part (a), I need to explicitly construct a topology relative to which are functions are continuous . But how do I this if not by constructing a collection of subbasic elements like that in part (b)?