Is the complement of a closed set always open? My book says that that a set is closed if its complement is open. Can a set be closed for other reason or is this if supposed to be an iff? Is there a way to prove this statement?
 A: By definition, a subset of a topological space is closed if and only if its complement is open.
A: As some of the other answers have said, in the field of topology, we define a set to be closed if it is the complement of some open set.  
The reason this makes sense as a definition is that for all the spaces we study in analysis, the closed sets (i.e., the sets which contain all their limit points) are precisely those sets that are the complements of open sets (i.e., sets theat contain an open ball around every point).  
The second statement clearly requires a proof.  I'll let you play around and try and find one for yourself, but yes - there is an 'iff' here - in any metric space $X$, a subset $K\subset X$ is closed if and only if $X\setminus K$ is open.  You know how to prove

$X\setminus K$ is open $\Rightarrow$ $K$ is closed.

Can you also prove

$K$ is closed $\Rightarrow$ $X\setminus K$ is open?

(You can also find this question asked here.)
The point is that when we come to generalize metric spaces to topological spaces, the notion of a set 'containing all its limit points' is no longer a good one, so we instead define a set to be closed if and only if it is the complement of an open set.  But we would never make that definition if it wasn't true for closed subsets of the metric spaces we study in analysis.  
A: This is simply a slightly confusing, but very widespread, convention of mathematical wording.
In statements, “if” is interpreted just as one direction of implication. To specify bidirectional implication, as you say, one needs to write “iff”, or “if and only if”, or “exactly if”, or similar.
In definitions, however, “if” is used for by-definition equivalence, which in particular gives the biconditional. (Quite arguably, it’s exactly the same as a biconditional; logicians can and do split hairs over this issue, but in standard foundations of mathematics, there’s essentially no difference.)
