I want to prove $A$ is closed iff $\overline{A}=A$;
I need to use the definition of neighbourhoods instead open sets and not use the complement to prove this.
So wondering how can you prove it just using closure I can't see how it is possible.
Edit it asks you to use the definition of closure directly. Suppose that X is a topological space and A is a subset of X. A point $x \in X$ is a closure points of A if for all neighbourhoods N of x, $N\cap A \not= \emptyset$. The set of closure points of A is the closure of A.