I am trying to learn, through self-study, about topological manifolds for the first time and would appreciate someone clarifying a point for me.
In proofs about topological manifolds there are frequent references made to subsets. Are the referenced subsets always one or more of the subsets that comprise the topology, or can they be special subsets that carve out new spaces in the topology but are not actual subsets of the topology?
For example, are balls, or subsets that are dense in a topology, always among those that constitute the topology?