As known from the works of Serra, Ronse, Haralick and Shapiro, the connectivity relations are found to be useful in filtering binary images. But it can be used also to find roadmaps in robot motion planning, i.e. to build discrete networks of simple paths connecting points in the robot's configuration space capturing the connectivity of this space. This paper generalizes and puts together the notion of a connectivity class introduced by Serra, and the notion of a separation relation. This gives an opportunity to introduce approximate epsilon-connectivity, and thus we show the relation between our approach and the Epsilon Geometry introduced by Guibas, Salesin and Stolfi. The duality between the notions 'connectivity class' and 'separation relation' has been established. As an application we consider the problem of cleaning drop-out noise from binary images by morphological closing filter. Ronse and Serra have defined connectivity analogs on complete lattices with certain properties. As a particular case of their work we consider the connectivity of fuzzy compact sets, which is a natural way to study the connectivity of gray-scale images. This idea can be transferred also in planning robot trajectories in the presence of uncertainties. Since based on fuzzy sets theory, our approach is intuitively closer to the classical set oriented approach, used for binary images and robot path planning in known environment with obstacles. This makes our theory much easier to implement, compare to the direct application of the beautiful and more general approach based on connectivity in functional spaces as presented.