We present three criteria for compactness in the context of apartness spaces and Bishop-style constructive mathematics. Each of our three criteria can be summarised as requiring that there is a positive distance between any two disjoint closed sets. Neat locatedness and the product apartness give us three variations on this theme. We investigate how our three criteria relate to one another and to several existing compactness criteria, namely classical compactness, completeness, total boundedness, the anti-Specker property, and Diener's neat compactness.

The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. In Paper I we focus on the notion of a domain representation of a formal space as a way to introduce generalized points of the represented space, whereas we in Paper II give a constructive and point-free treatment of the domain theoretic approach to differential calculus. The last two papers are of a slightly different nature but still concern constructive topology. In paper III we consider a measure theoretic covering theorem from various constructive angles in both point-set and point-free topology. We prove a point-free version of the theorem. In Paper IV we deal with issues of impredicativity in the theory of apartness spaces. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces.