Smallness in topology

Abstract

Quillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy  theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological  spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many  familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant  differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition  standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective  subcategories of the category of all topological spaces.