Compactness and the Stone-Weierstrass theorem in pointfree topology

Pointfree topology is, as the name suggests, a way of studying spaces without (mentioning) points. This idea is more natural than one might initially think. For example, when drawing a point on paper, we do no draw an actual point, but a collection of points somewhere near the desired one. We drew a “spot”, which can be reduced in size if that would be required to serve our purposes. Hence it makes sense not to use points anyway. Moreover, pointfree topology allows us to obtain similar and sometimes even better results than in classical topology. It is for instance possible to make a fully constructive Čech-Stone compactification, where the classical one is heavily dependent on the axiom of choice. Even more surprising is the fact that the pointfree Tychonoff Theorem can also be proven constructively, whereas the classical one is equivalent to the axiom of choice. One might also think that forgetting points means that we lose information. We will show that no information will be lost if we assume some sort of separation. In this thesis, we will first define and study frames, which are pointfree counterparts of topological space. We will show that they form a proper generalisation of spaces and a way of reconstructing spaces will be given. Moreover, we will study useful properties including free frames, subspaces, categorical limits and separation axioms Finally, we will study some compactness properties in frames. A ČechStone compactification using only countable dependent choice will be constructed and as a corollary, part of the Tychonoff Theorem will follow. Then the pointfree counterpart to the real numbers will be introduced to allow the proof of the pointfree Stone-Weierstrass Theorem.