I agree largely. However, be warned that sometimes there are several ways to define things, which are sometimes not equivalent.
For example, in metric spaces, we can define continuity using $\varepsilon$-$\delta$ balls, and we can show this definition to be equivalent with one in terms of open sets (the latter would be a proposition, mind you). When we enter the realm of topology, we cannot speak of distance anymore, and we define continuity in terms of open sets. It's not true that a definition in terms of $\varepsilon$-$\delta$ balls is or is not equivalent - it just does not mean anything!
Therefore, the fact that even though some function on a topological space may be continuous (as defined in terms of open sets), this does not always imply that we can speak of an "open ball" (which we would conclude from the metric definition of continuity).