245B, Notes 13: Compactification and metrisation (optional)

One way to study a general class of mathematical objects is to embed them into a more structured class of mathematical objects; for instance, one could study manifolds by embedding them into Euclidean spaces. In these (optional) notes we study two (related) embedding theorems for topological spaces:

• The Stone-Čech compactification, which embeds locally compact Hausdorff spaces into compact Hausdorff spaces in a “universal” fashion; and
• The Urysohn metrization theorem, that shows that every second-countable normal Hausdorff space is metrizable.