Abstract
Noiri and Popa introduced and studied the notion of M−closed function between spaces with minimal structures, developing a unified theory of modifications of closedness such as α−closedness, semi-closedness, preclosedness and β−closedness. Using a new notion, that of almost M- closed function, we extend the characterizations of M−closed functions proved by Popa and Noiri to spaces endowed with minimal structures not necessarily closed under arbitrary unions. These minimal structures are useful beyond General Topol-ogy. For bijections between spaces with minimal structure it turns out that almost M- closedness is equivalent to almost M−openness, both being equivalent to the M−continuity of the inverse function. Our main result generalizes a well-known theorem of Long and Herrington showing that θ−open sets in topological spaces are preserved by every function that is both open and closed.