When Is One Compactification Larger Than Another?
A compactification adds points to a space so that the enlarged space is compact. Different compactifications can add different points or organize the added points differently. A useful way to compare them is to ask whether one can be mapped continuously onto another without changing any point of the original space. The direction of this map matters: a compactification that can map onto another retains at least as much information about how the original space approaches its boundary.
In this tutorial, a compactification of a Hausdorff space \(X\) means a compact Hausdorff space \(K\) together with an embedding \(e:X\to K\) whose image is dense in \(K\). We often identify \(X\) with its embedded image. Two compactifications are equivalent if there is a homeomorphism between them that fixes \(X\) pointwise. We use \(\beta X\) for the Stone–Čech compactification of a completely regular Hausdorff space \(X\), and \(\alpha X\) for the one-point compactification of a locally compact, noncompact Hausdorff space \(X\).
We will use two basic facts about compactifications. First, a continuous map \(f:\gamma X\to\alpha X\) that fixes \(X\) must be onto: its image is compact, hence closed in the Hausdorff space \(\alpha X\), and it contains the dense subset \(e_\alpha(X)\). Second, any two continuous maps from a space into a Hausdorff space that agree on a dense subset agree everywhere. Indeed, the set where the maps agree is closed: if their values differ at a point, disjoint neighborhoods of those values give a neighborhood on which they still differ. This closed set contains the dense subset, so it is the whole domain.
The Order on Compactifications
Proof. The identity map on any compactification fixes \(X\), so \(\alpha X\preceq\alpha X\). If \(f:\gamma X\to\alpha X\) fixes \(X\) and \(g:\delta X\to\gamma X\) fixes \(X\), then \(f\circ g:\delta X\to\alpha X\) is continuous and, for every \(x\in X\), satisfies \((f\circ g)(e_\delta(x))=e_\alpha(x)\). Thus the relation is transitive. Now suppose there are maps \(f:\gamma X\to\alpha X\) and \(g:\alpha X\to\gamma X\), both fixing \(X\). The composite \(f\circ g\) agrees with the identity on the dense copy of \(X\) in \(\alpha X\), so the uniqueness fact above implies \(f\circ g=\operatorname{id}_{\alpha X}\). Likewise, \(g\circ f=\operatorname{id}_{\gamma X}\). Therefore \(f\) and \(g\) are inverse homeomorphisms fixing \(X\). This proves antisymmetry after equivalent compactifications are identified. \(\square\)
The order records more than the number of added points. A map fixing \(X\) may identify several boundary points, but it cannot alter any point of \(X\). The compactification at the source of the map is therefore the one that can be collapsed to obtain the target.
Stone–Čech Maps onto the One-Point Compactification
The Stone–Čech compactification has the following universal property: if \(X\) is completely regular and Hausdorff and \(K\) is compact Hausdorff, every continuous map \(h:X\to K\) extends to a continuous map \(\overline{h}:\beta X\to K\). We use this standard property, established in the earlier material on Stone–Čech compactification.
Proof. Let \(e_\alpha:X\to\alpha X\) be the embedding into the one-point compactification. The space \(\alpha X\) is compact Hausdorff, so the universal property extends \(e_\alpha\) to a continuous map \(F:\beta X\to\alpha X\). By construction, \(F\) fixes \(X\). The image \(F(\beta X)\) is compact because \(\beta X\) is compact. It is closed because \(\alpha X\) is Hausdorff. It contains \(e_\alpha(X)\), which is dense in \(\alpha X\). A closed set containing a dense subset is the whole space, so \(F(\beta X)=\alpha X\). Hence \(F\) is onto and \(\alpha X\preceq\beta X\). \(\square\)
Worked Example: The Stone–Čech Compactification of the Real Line
Let \(X=\mathbb{R}\) with its standard topology. It is noncompact, locally compact, and completely regular Hausdorff, so the theorem gives a continuous surjection \(F:\beta\mathbb{R}\to\alpha\mathbb{R}\) fixing \(\mathbb{R}\). The one-point compactification \(\alpha\mathbb{R}\) is homeomorphic to the circle \(S^1\).
To specify the circle embedding, set \(t(x)=\frac12+\frac{\arctan x}{\pi}\), which lies in \((0,1)\), and send \(x\) to \(e^{2\pi i t(x)}\). As \(x\) ranges over \(\mathbb{R}\), \(t(x)\) ranges over \((0,1)\), and this formula covers the circle except for \(1\). In the one-point compactification the missing point is the point at infinity. Thus the map \(F\) sends each point of \(\mathbb{R}\subseteq\beta\mathbb{R}\) to its specified circle point, and sends the rest of \(\beta\mathbb{R}\) onto the missing point. To check the last statement, a point outside \(\mathbb{R}\) cannot map to a point \(x\in\mathbb{R}\): the singleton \(\{x\}\) is open in \(\alpha\mathbb{R}\), so its preimage would be an open neighborhood of that point disjoint from the dense copy of \(\mathbb{R}\), which is impossible.
The fibers over points of \(\mathbb{R}\) are singletons by the same argument: the preimage of \(\{x\}\) is open, and if it contained a point other than \(x\), an open neighborhood of that point could be chosen to avoid \(x\); this would give a nonempty open subset of \(\beta\mathbb{R}\) missing the dense copy of \(\mathbb{R}\). All points of \(\beta\mathbb{R}\setminus\mathbb{R}\) therefore map to the point at infinity. This illustrates how a large remainder can be collapsed to a single point.
Worked Example: The Stone–Čech and One-Point Compactifications of \(\mathbb{N}\)
Give \(\mathbb{N}=\{1,2,\ldots\}\) the discrete topology. Its one-point compactification \(\alpha\mathbb{N}=\mathbb{N}\cup\{\infty\}\) has the original points isolated, and a neighborhood of \(\infty\) contains all but finitely many natural numbers. The inclusion of \(\mathbb{N}\) into \(\alpha\mathbb{N}\) extends by the Stone–Čech universal property to a continuous surjection \(F:\beta\mathbb{N}\to\alpha\mathbb{N}\). It fixes every \(n\in\mathbb{N}\).
Every point of \(\beta\mathbb{N}\setminus\mathbb{N}\) maps to \(\infty\). For if such a point mapped to \(n\), the open set \(F^{-1}(\{n\})\) would contain it. Since \(\beta\mathbb{N}\) is Hausdorff, this open set has a nonempty open subset avoiding \(n\). That subset cannot meet \(\mathbb{N}\), because the only natural number mapping to \(n\) is \(n\) itself. This contradicts the density of \(\mathbb{N}\) in \(\beta\mathbb{N}\). As in the real-line example, the fibers over natural numbers are singletons, and the entire remainder maps to \(\infty\).
There is no continuous map \(\alpha\mathbb{N}\to\beta\mathbb{N}\) fixing \(\mathbb{N}\). To see this, consider the sequence \(1,2,3,\ldots\). It converges to \(\infty\) in \(\alpha\mathbb{N}\), because for every finite set of natural numbers all sufficiently late terms lie outside that set. If such a map existed, continuity would force this sequence to converge in \(\beta\mathbb{N}\) to the image of \(\infty\). But no limit exists: the even and odd natural numbers have disjoint clopen closures in \(\beta\mathbb{N}\), and the sequence has infinitely many terms in each. A convergent sequence must eventually lie in every neighborhood of its limit, including whichever of those two clopen sets contains the limit. It cannot do so for both sets. Thus \(\alpha\mathbb{N}\preceq\beta\mathbb{N}\), but not \(\beta\mathbb{N}\preceq\alpha\mathbb{N}\).
Comparing Two Compactifications of the Real Line
The direction of the order can also be seen by comparing the one-point compactification of \(\mathbb{R}\) with a compactification that retains two different directions of escape. Write \([-\infty,+\infty]\) for the extended real line with its order topology. Embed \(\mathbb{R}\) in it in the usual way. The result is compact Hausdorff, and \(\mathbb{R}\) is dense: every neighborhood of either endpoint contains real numbers. This gives a compactification with two added endpoints.
Worked Example: Collapsing Two Ends to One
We compare \(\alpha\mathbb{R}\) and \([-\infty,+\infty]\). There is a continuous map \(q:[-\infty,+\infty]\to\alpha\mathbb{R}\) fixing \(\mathbb{R}\) and sending both endpoints to the single point at infinity. For an explicit description, use \(t(x)=\frac12+\frac{\arctan x}{\pi}\) on finite \(x\), with \(t(-\infty)=0\) and \(t(+\infty)=1\). Map \(t\in[0,1]\) to \(e^{2\pi i t}\). This is continuous, takes both endpoints to \(1\), and restricts to the chosen circle embedding of \(\mathbb{R}\). Since \(\alpha\mathbb{R}\cong S^1\), it gives the required map. Therefore \(\alpha\mathbb{R}\preceq[-\infty,+\infty]\).
The reverse map cannot fix \(\mathbb{R}\). In \(\alpha\mathbb{R}\), both sequences \(n\) and \(-n\), for \(n\in\mathbb{N}\), converge to the same point \(\infty\): every compact subset of \(\mathbb{R}\) is bounded, so each sequence is eventually outside it. In \([-\infty,+\infty]\), the sequence \(n\) converges to \(+\infty\), while \(-n\) converges to \(-\infty\). If a continuous map \(\alpha\mathbb{R}\to[-\infty,+\infty]\) fixed \(\mathbb{R}\), continuity would make both image sequences converge to the image of \(\infty\). Limits in a Hausdorff space are unique, but \(+\infty\ne-\infty\). This contradiction rules out the reverse map.
This example shows what the order detects: the two-ended compactification remembers which direction a point escapes in, while the one-point compactification identifies both directions.
A Common Mistake About the Direction
Common pitfall. It is easy to reverse the meaning of \(\preceq\) and say that \(\alpha X\preceq\gamma X\) requires a map \(\alpha X\to\gamma X\). That is not our convention. For example, the map \([-\infty,+\infty]\to\alpha\mathbb{R}\) collapses the two endpoints, so the correct statement is \(\alpha\mathbb{R}\preceq[-\infty,+\infty]\). The failed reverse map confirms that changing the arrow changes the claim, not just its notation.
When comparing compactifications, first write the arrow and the condition that it fixes \(X\). Then check whether the map exists and, if so, whether it is onto. Compactness of the source and Hausdorffness of the target make surjectivity automatic once the map fixes a dense copy of \(X\). The Stone–Čech universal property supplies such maps to compact Hausdorff targets; the one-point compactification is one important target when \(X\) is locally compact.
Check Your Understanding
Use the definition of \(\preceq\) and the examples above to check your reasoning.
- Write the direction of the map required for \(\alpha X\preceq\gamma X\), and state why this map is onto.
- Why does the Stone–Čech universal property give a map \(\beta X\to\alpha X\) under the hypotheses of the theorem?
- In the comparison of \(\alpha\mathbb{R}\) with \([-\infty,+\infty]\), which compactification remembers the two directions of escape?
- Why can there be no continuous map \(\alpha\mathbb{R}\to[-\infty,+\infty]\) fixing \(\mathbb{R}\)?
- Explain why maps in both directions that fix \(X\) imply that the two compactifications are equivalent.