1. ## iff statement logic

I'm working to prove this statement, but the wording is throwing me off.

"A set W is closed iff for each x, if every neighborhood of x intersects W, then x in W."

This statement is talking about a larger topology X containing W, and I assume x is just an element in X.

-----> Assume W is closed and that x is an element for which every neighborhood intersects W, then show x is in W
<----- Assume for some point x, every neighborhood of x intersects W implies x is in W, then show W is closed

Would this be the correct approach? These sort of multiple if statements throw me off, any good articles on dealing with them? Thanks.

2. Yes thats right. When proving $\iff$ statements you will almost always want to break the proof down into the forward and backward directions. Keep in mind that sometimes when you are proving the backward or forward direction that the obvious way to proceed might be impossible but the contrapositive may be easy to prove. That is, suppose you are trying to prove $a \iff b$ then this is equivalent to $(a \implies b) \wedge (b \implies a)$ which is equivalent to $(\neg b \implies \neg a) \wedge (b \implies a)$ which is equivalent to $(a \implies b) \wedge (\neg a \implies \neg b)$ which is equivalent to $(\neg a \implies \neg b) \wedge (\neg b \implies \neg a)$