Connected operators from mathematical morphology to some extent model perceptual grouping in human vision. However, classical connectivity has several drawbacks, in particular its inability to deal with overlap. In this work, connected operators are extended to a wider class of operators, which are based on connectivities in higher-dimensional spaces, similar to scale spaces, which will be called attribute spaces. Though some properties of connected filters are lost, granulometries can be defined under certain conditions, and pattern spectra in most cases. The advantage of this approach is that regions can be split into constituent parts before filtering more naturally than by using partitioning connectivities. Furthermore, the approach allows dealing with overlap, which is impossible in connectivity. The theoretical results are illustrated by several synthetic examples, and briefly in an application to neuron images.