What is compactification generally?

Question

In wikipedia, compactification is defined as an topological imbedding $f:X\rightarrow Y$ such that $f(X)$ is dense in $Y$.

However, Munkres-Topology requires $Y$ to be Hausdorff to be called a compactification.

For example, let $X$ be locally compact Hausdorff which is noncompact. Then, there exists a compact Hausdorff space $Y$ such that $X$ is dense in $Y$ and $Y\setminus X$ is a singleton.

What would be more general and natural to call this space? One-point Hausdorff compactification? or One-point compactification?

Answer 1

In defining the topology on the one point compactification, one needs to claim that $X\setminus K \cup \{\infty\}$ be open. But if $X$ is not Hausdorff, $K$ might not be closed. Thus it might be more natural to assume that $X$ is Hausdorff.

There are a lot of compactification, as long as $f(X)$ is dense in $Y$. For example, you can do either a one point or two point compactification of $(0, 1)$, to make it into a circle or a close interval. You can also compactify the open unit disc by one point compactification, which becomes a sphere, or you can compactify by adding the boundary circle.

Which one you choose depends on what you want to achieve.

Answer 2

If you start with a Hausdorff space $X$ and want to consider a compactification, which by definition is a pair $(Y, f)$ where $Y$ is a compact space (let's be the most general and not require Hausdorff) and $f: X \to Y$ is an embedding and $f[X]$ is dense in $Y$.

Trivial example: if $X$ is already compact (and not even Hausdorff) then $(X, 1_X)$, where $1_X$ is the identity on $X$, is a compactification.

Also, for any space $X$, define $Y = X \cup \{p\}$, with $p \notin X$, with the topology defined by $\mathcal{T}_X \cup\{{Y}\}$ (the original topology on $X$ with just the whole set $Y$ added, which indeed is a topology on $Y$, as is easily checked), and let $f(x) = x$ be the obvious map from $X$ into $Y$. Then $f$ is an embedding, the only neighborhood of $p$ is $Y$ and this implies both that $f[X]$ is dense in $Y$ and that $Y$ is compact as well. In fact we can add more than just one point, we can add as many as we like... The resulting spaces are all compact for trivial reasons, and even if we start with a nice metrisable space, the compactifications are at most $T_1$ and never $T_2$. So allowing all spaces $Y$ allows for all sorts of trivialities.

This is the reason that most texts assume that $Y$ is Hausdorff as well, and so even $T_4$ by standard results. Note then that $X$, being homeomorphic to $f[X] \subset Y$ is at least required to be Tychonoff (a.k.a. $T_{3\frac{1}{2}}$) itself in that case, and because every Tychonoff space embeds into some product $[0,1]^I$, it always has a compactification (take $Y$ the closure of the embedded copy of $X$, essentially). So allowing only Hausdorff only compactifications, we get that we restrict to Tychonoff spaces, all of which actually have Tychonoff (even normal) compactifications, so we stay inside the same class of spaces, as opposed to the trivial example spaces above.

The one-point compactification case then becomes a bit clearer too: if we just take any space $X$ and call $(Y,f)$ a one-point compactification when $Y \setminus f[X]$ has one point, then a (even Hausdorff locally compact) space $X$ can have more then one non-homeomorphic one-point compactification: $\mathbb{R}$ has the circle, as usual, and its trivial extension from above, and more as well.

If $X$ is Hausdorff and we assume that all compactifications are Hausdorff, we get as a theorem that a one-point Hausdorff compactification $Y$ exists iff $X$ is locally compact and then it is essentially unique as well (as all such $Y$ must be homeomorphic).

If $X$ is just Hausdorff, we can define a space $Y = X \cup \{p\}$ (where $p \notin X$) with the topology $$\mathcal{T}_X \cup \left\{ \{p\} \cup (X \setminus K): K \mbox{ compact and closed in } X \right\}$$ and this space is such that the $Y$ is indeed compact but it needs not be Hausdorff: in fact this space is Hausdorff only if $X$ was locally compact to begin with. If however $X$ is locally compact, it is the same unique Hausdorff compactification as before, so this space is the natural generalisation of the Hausdorff compactification. But as said, it's not unique outside of the Hausdorff setting, though it is IMHO the most natural one.