Alternative ways to say “if and only if”?
So when I come across mathematical definitions like “A function is continuous if….”A space is compact if….”,”Two continuous functions are homotopic if…..”, etc when is it okay to assume that the definition includes the converse as well?
Absolutely. The definition will state that we say [something] is $P$ if $Q$. Thus, every time that $Q$ holds, $P$ also holds. The definition would be useless if the other direction didn’t hold, though. We want our terms to be consistent, so it is tacitly assumed that we will also say $P$ only if $Q$. Many texts prefer to avoid leaving this as tacit, and simply state it as “if and only if” in their definitions.
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).