Abstract: To study topology in digital binary images, different discrete connectivities have to be used for both the object and the background. We prove equivalences between discrete connectivities, as they are usually defined in a digital space and connectivity in a continuous space. For that purpose, we introduce the notion of regularization of a set of voxels. It will permit us to define continuous sets from discrete ones. We discuss then the choice of a good couple of discrete connectivities for both the object and the background and provide a characterization of such a valid pair. Finally, we use our best choice to demonstrate a separation theorem in dimension n for surfaces made of voxels.