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.