Compactness

In this video, we look at a topological property called compactness. Compact spaces are extremely important in mathematics because they generalise, in a certain sense, the notion of finiteness. They are thus often much better behaved than non-compact spaces. In fact, a common technique for studying non-compact spaces is to "compactify" them, that is, embed them in a compact space. For example, in algebraic geometry, we compactify affine space to obtain projective space.

In this video, we define the notion of compactness. We then look at the Heine-Borel theorem, which characterises compact subsets of Euclidean space as precisely those which are closed and bounded. This is rather surprising because boundedness cannot be characterised topologically. We also examine some basic facts about compactness.