With respect to a closure operator C in a topological category X, subcategories of X are defined by using C in terms of separation axioms such as T_0 and T_1. Then the
epis are found these categories. By taking a particular closure operator K, results about epis and co-wellpoweredness are obtained in case X=Fil, Lim, PsT, PrT, some of which are not true when X is the category of all topological spaces.
