We recently introduced the topological tree of shapes as a new structure in the family of morphological trees. The topological tree of shapes is both a topological invariant of grey-level images and a hierarchical model of these images. Building upon these two properties, we explain how the topological tree of shapes can be used to relevantly design connected operators. In this theoretical article, we show that such operators present algebraic properties related to openings/closings, while also guaranteeing the preservation of the topological structure of the processed images. Software implementation available at https://github.com/jmendesf/TopologicalTreeOfShapes .