Space Connectivity and Translation-Invariance

Abstract

This paper shows that the space connectivity (defined by γ _x ) must satisfy certain condition so that some intuitively expected behavior is guaranteed, such as, for example, to guarantee that a translated connected component is actually a connected component. This is a major issue when connected operators are employed. This work proposes a condition that should be satisfied by the opening γ _x in order to avoid the appearance of counter-intuition results in spaces equipped with translation. In particular, this study has investigated the relationship between γ _x with openings and closings by reconstruction when translation invariance is desired. An important result is the fact that it can be impossible to build translation invariant openings and closings by reconstruction if γ _x does not satisfy the proposed condition.