I'm currently learning about compact spaces, apparently they are useful in lots of areas of Mathematics, but I can't find any references to this online? Can anyone give me more details to where they arise and why they are useful?
Note: I'm learning about them in a topological sense, using Munkres Topology.
This is a really simple application of the notion of compact spaces. Consider $\mathbb{R}$ as a metric space with the Euclidean metric. Consider any closed interval, $[a,b]\subset\mathbb{R}.$ From the Heine-Borel theorem, we know that $[a,b]$ is compact. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Because we know the continuous image of a compact is compact, we know that $f([a,b])$ is also compact.
So, we know that the image of a continuous function on a closed and bounded subset of $\mathbb{R}$ must attain a maximum and a minimum on that interval. Of course, this is a simple example, but this is a simple proof of a powerful result courtesy of our knowledge of compactness.