In this paper a new class of sets, namely $\psi-\mathbf{closed\: set}$
is introduced for topological spaces. This class falls strictly in
between the class of semi-closed sets and the class of semipreclosed
sets. This class also sits strictly in between the class of semi-closed
sets and the class of semi-generalized closed sets. We also introduce
and study a new class of spaces, namely $\mathbf{semi-}\textrm{T}_{1/3}\:\mathbf{spaces}$.
Further we introduce and study $\psi-\mathbf{continuous}$ maps and
$\psi-\mathbf{irresolute}$ maps.
