I'm heading into my second semester of analysis, and I still don't have a good intuition of when a set is compact. I know two definitions, covering compact and sequentially compact, but both of those seem very difficult to apply "real time". In $\mathbb{R}^n$ I know we have Heine-Borel, which is a very easy way to check things, but I would like to know a different way to easily check for compactness, if there is one.
For instance, can you compactify any space? Compactifying Euclidean Space is easy to understand because of Heine-Borel, but can you compactify, for instance, $\mathbb{R}^2$ with the discrete metric?