# 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?

• Possibly relevant: pseudocompactness does not imply compactness. A pseudocompact space $X$ is one where every continuous real-valued function on $X$ has a bounded range. The Heine-Borel theorem is a sort of converse to this.
– MJD
Jun 25, 2014 at 20:19
• @MJD Compactness in $\mathbb R$ is equivalent to pseudocompactness, so Heine Borel says closed and bounded iff pseudocompact. Jun 25, 2014 at 20:22

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 $S^2$.

• Artificial and outrageous but very cool! Jun 25, 2014 at 19:49
• Why is it locally compact? And @TomCruise, what is very cool- the Stone-Cech compactification?
– user82004
Jun 25, 2014 at 19:54
• @Anthony Locally, a discrete space is finite, and finite spaces are compact. Stone–Čech is very interesting in logic, and when doing universal constructions, but in my opinion it is not very geometric. Jun 25, 2014 at 20:05
• @Anthony Yes, you can find many interesting combinations of topological properties by studying them. Although they are not so applicable to analysis... the Stone-Cech compactifications of $\mathbb N$ and $\mathbb R$ are not even metrizable. Jun 25, 2014 at 20:07
• One more quick question, what is a metric (or a topology?) where closed and bounded does not imply compact?
– user82004
Jun 25, 2014 at 20:09

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

• There are usually conditions on $G$ that are required for it to be suitable for one-point compactification. My memory says is should be locally compact, noncompact, and Hausdorff. I don't remember if these are actually required, but I think at least the local compactness is required.
– MJD
Jun 25, 2014 at 20:16
• @MJD for $\mathbb{R}^2$ under the discrete metric, $\{x\}$ is an compact neighborhood of $x$ for every $x \in \mathbb{R}^2$, so we do have local compactness. That the other two properties hold should be clear. Jun 25, 2014 at 20:18
• I am not quibbling with your construction for the compactification of the discrete topology, only with your claim that “it is possible to compactify any space. An easy way to do this is [one-point compactification]”. This is misleading unless is is actually possible to use the one-point compactification to compactify any space.
– MJD
Jun 25, 2014 at 20:20
• @MJD oh, I see. Fair point. Jun 25, 2014 at 20:20
• @MJD I think those are only required if we specify that the one-point compactification should be Hausdorff, which is not universally the practice. Jun 25, 2014 at 20:23

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.