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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.
I’m continuing my series of articles for the Princeton Companion to Mathematics with my article on compactness and compactification. This is a fairly recent article for the PCM, which is now at the stage in which most of the specialised articles have been written, and now it is the general articles on topics such as compactness which are being finished up. The topic of this article is self-explanatory; it is a brief and non-technical introduction as to the incredibly useful concept of compactness in topology, analysis, geometry, and other areas mathematics, and the closely related concept of a compactification, which allows one to rigorously take limits of what would otherwise be divergent sequences.
The PCM has an extremely broad scope, covering not just mathematics itself, but the context that mathematics is placed in. To illustrate this, I will mention Michael Harris‘s essay for the Companion, ““Why mathematics?”, you may ask“.