The authors consider the notions of near compactness, near cocompactness, near stability, near costability and near dicompactness in the setting of ditopological texture spaces. In particular preservation of these properties under surjective R-dimaps, co-R-dimaps and bi-R-dimaps is investigated and non-trivial characterizations of near dicompactness are given which generalize those for dicompactness. The notions of semiregularization, semicoregularization and semibiregularization are defined and used to give generalizations of Mrówka’s Theorem for near compactness and near cocompactness, and of Tychonoff’s Theorem for near compactness, near cocompactness and near dicompactness. Also, results related to pseudo-open and pseudo-closed sets are presented. Finally, examples are given of co-$T_{1}$ nearly dicompact ditopologies on textures which are not nearly plain and an open question is posed.