Intuition Behind Compactification 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? 

A compactification of a Hausdorff space $X$ is a compact Hausdorff $\alpha X$ containing a dense copy of $X$, i.e., cl$_{\alpha X} X=\alpha X$. So for instance $[0,1]$ is a compactification of $(0,1)$.
I think the general idea is you want to embed (put a copy) a space into a compact space because compact spaces are nice to work with.  If you have a dense embedding of $X$ into a compact space $\alpha X$, then in a sense $\alpha X$ closely resembles $X$.  To get existence results we usually require $X$ and $\alpha X$ to be at least Hausdorff.  
It may seem counterintuitive to get a compact space by adding points to a non-compact space.  The important thing for compactness is not the underlying set, but rather the topology you place on this set extension.
Heine-Borel is a very effective tool to check for compactness in $\mathbb R ^n$.  Usually boundedness is easy to determine.  If the set is bounded, it boils down to showing it is  closed.
