Homotopical smallness and closeness

The aim of this paper is to introduce the concepts of homotopical smallness and closeness. These are the properties of homotopical classes of maps that are related to recent developments in homotopy theory and to the construction of universal covering spaces for non-semi-locally simply connected spaces, in particular to the properties of being homotopically Hausdorff and homotopically path Hausdorff. The definitions of notions in question and their role in homotopy theory are supplemented by examples, extensional classifications, universal constructions and known applications.