The author introduces a new class of generalized closed sets which he calls \(\psi\)-closed sets. A subset \(A\) of a topological space \(X\) is said to be \(\psi\)-closed if \(\text{scl}(A) \subseteq U\) whenever \(A\subseteq U\) and \(U\) is sg-open in \(X\). With the help of \(\psi\)-closed sets, the author is able to provide new characterizations of semi-regular sets, regular open sets and clopen sets. In addition, \(\psi\)-closed sets give rise to a new separation axiom called semi-\(T_{1/3}\), as well as new types of generalized continuous maps, namely \(\psi\)-continuous maps and \(\psi\)-irresolute maps. These concepts are investigated and compared to other notions of generalized continuity.