Varying compactness

Introduction
Compactness is one of the most useful topological concepts that one sees throughout point-set topology, algebraic topology, analysis, algebraic geometry, logic and model theory. Variations of compactness, in different veins, have been studied by a wide variety of mathematicians. A reasonably complete list of variations of compactness is available at:

Category:Variations of compactness

This article tries to explore important variations of compactness, organizing them into themes and streams. The key idea will be to understand how and why the property of compactness is useful, and how we can emulate these strengths by lesser properties (almost all the variations we study here will be weakenings).