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? 

There's a few different questions going on here, but I'll focus on the last one:
Yes, it is possible to compactify any space.  An easy way to do so is to take your space $X$ and add a point called "$\infty$", and we say that a set $G$ containing $\infty$ is open if and only if $(X \cup \{\infty\}) \setminus G$ is a compact set in $X$ (and therefore in $X \cup \{\infty\}$).  This is called the "one-point compactification".
So, taking your example: the only compact sets in $\mathbb{R}^2$ under the discrete metric are those with finitely many points.  So, we can take the compactification $\mathbb{R}^2 \cup \{\infty\}$ by saying that we only call a set containing $\infty$ open if it contains all but finitely many of the elements of $\mathbb{R}^2$.  Notice that this is very much distinct from the discrete topology on $\mathbb{R}^2 \cup \{\infty\}$.
