The notion of an almost fully closed mapping, which generalizes the definition of a fully closed map, is introduced. It is shown that the composition of almost fully closed mappings of compacta with the first axiom of countability is also almost fully closed, and so is the limit of a countable inverse transfinite sequence of the same mappings. A class of quasi-$F$-compacta (which contains the class of Fedorchukcompacta) is defined. For this class, some statements are proved that generalize and strengthen the known theorems on $F$-compacta.