The notion of connectivity is very important in image processing and analysis, and particularly in problems related to image segmentation. It is well understood, however, that classical notions of connectivity, including topological and graph-theoretic notions, are not compatible with each other. This motivated G. Matheron and J. Serra to develop a general framework of connectivity, which unifies most classical notions, circumvents incompatibility issues, and allows the construction of new types of connectivity for binary and grayscale images. In this paper, we enrich this theory of connectivity by providing several new theoretical results and examples that are useful in image processing and analysis. In particular, we provide new results on the semi-continuity behavior of connectivity openings, we study the reconstruction operator in a complete lattice framework, and we extend some known binary results regarding reconstruction to this framework. Moreover, we study connectivities constructed by expanding given connectivities by means of clustering operators and connectivities constructed by restricting given connectivities by means of contraction operators.