Although a nontopological concept, boundedness seems to be of considerable importance in a topological space. 'There are many topological problems in which it is essential to be able to make this distinction' (between bounded and unbounded sets) . Boundedness and in particular boundedness-preserving' uniform spaces appear to have applications to topological dynamics .
In spite of this importance, there have been only isolated attempts at developing the concept. Alexander  and Hu  tried the axiomatic approach. Hu, for example, calls a nonempty family of sets a boundedness if is hereditary and closed under finite union.