Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In this context, we present an introductory study of the new notion of simple set which extends the classical notion of simple point. Similarly to simple points, simple sets have the property that the homotopy type of the object in which they lie is not changed when such sets are removed. Simple sets are studied in the framework of cubical complexes which enables, in particular, to model the topology in Z^n. The main contributions of this article are: a justification of the study of simple sets (motivated by the limitations of simple points); a definition of simple sets and of a subfamily of them called minimal simple sets; the presentation of general properties of (minimal) simple sets in n-D spaces, and of more specific properties related to "small dimensions" (these properties being devoted to be further involved in studies of simple sets in 2; 3 and 4-D spaces).