# Can mathematical definitions of the form “P if Q” be interpreted as “P if and only if Q”? [duplicate]

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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?

## marked as duplicate by TonyK, user31373, Chris Eagle, t.b., Asaf Karagila♦Aug 17 '12 at 15:13

• Yes, definitions are "if and only if" statements. The "only if" part holds trivial content though, because it is defined like that. – Ragib Zaman Jul 10 '12 at 18:56
• In fact, definitions should be interpreted that way. Personally, following the lead of my favorite undergraduate professor, I always use the "if and only if" format, and when I use it in class, I make the (verbal) point of saying that it is an "if and only if" because it is a definition. – Arturo Magidin Jul 10 '12 at 23:32

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.
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!