Compactness in metric spaces

Let me start this post by quoting Adam Bobrowski:

I believe saying that the notion of compactness is one of the most important ones in topology and the whole of mathematics is not an exaggeration.

Agreed. Now let me add a quotation of my own:

I believe saying that metric spaces are the most important spaces in topology is not an exaggeration.

There are, of course, non-metrizable topological spaces, and they have their uses, but let's face it: those uses are rare and limited. I, for one, don't have a clue what they may be. Point-wise convergence of functions? Who gives an $\epsilon$?


So let's combine the two and investigate "metric compactness," by which I mean compactness of metric spaces. Our menu for today will be centered around two main dishes:

• The Heine-Borel theorem
• The Bolzano-Weierstraß theorem

Both are generalized into their astronaut-food equivalent, and some related concepts will be discussed.


Let's start with the related concepts. There are many different kinds of compactness,