Intuition Behind Compactification

Question

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?

Answer 1

Heine-Borel is probably close to the best intuition that one can get, at least for Hausdorff spaces. Compact spaces have a closed, finite character to them.

There are general ways to compactify any topological space, but many of these compactifications are very large and artificial. The Stone–Čech compactification is the most general and most outrageous.

For $X=\mathbb{R}^2$ with the discrete topology, we have a locally compact Hausdorff space, so we can use the one-point compactification. That is, we add a point "at infinity", whose open neighborhoods each contain all but finitely many points of $X$. This example is not very easy to visualize, but it is philosophically very similar to adding a point to $\mathbb{R}^2$ to obtain $S^2$.

Answer 2

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 $\mathbb{R}^2 \cup \{\infty\}$.

Answer 3

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.