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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.