Linear openness of multifunctions in metric spaces

Here, S stands for the closure of the set S. Various openness and almost openness notions are build on the inclusions (1.1) and (1.2). Every openness property implies the corresponding almost openness property, for S⊆ S. It is expected for the converse implication to be true in case that the multifunction F has a closed graph. In this regard, we recall the result in [12, Chapter 6, Lemma 36, page 202] (see also [7, Lemma 6.4.1, page 435]), where Y is a more general space, namely, a uniform space. The metric version of this result revolves around the composite inclusion