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Adding one symbol called infinity is easy. Choosing its neighbourhoods so the enlarged space is compact—and knowing when it remains Hausdorff—is the real construction.
Open the chapter contents · Topology root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Construct the extra point
Route 2 · Work concrete examples
Route 3 · Use local compactness
Route 4 · Test maps and limits
Route 5 · Check and continue
CHAPTER 1 OF 12 · CONSTRUCT THE EXTRA POINT
1. Adding a point is easy; choosing its neighbourhoods is not
Let X be a noncompact locally compact Hausdorff space. Its one-point compactification is X*=X∪{∞}. Every open set of X remains open in X*. A set containing ∞ is open when its complement in X is compact and closed. Equivalently, a basic neighbourhood of infinity has the form {∞}∪(X\K) for compact K⊂X.
This definition makes “going to infinity” mean eventually leaving every compact subset of X. The added point is not a largest coordinate and need not represent one geometric direction. It collects every way of escaping compact sets.
Local compactness is what lets ordinary points remain separable from ∞; Hausdorffness controls the compact-set behaviour used in that separation.
CHAPTER 2 OF 12 · CONSTRUCT THE EXTRA POINT
2. Compactness follows from one finite remainder
Take any open cover of X*. One member U contains ∞. Its complement K=X*\U lies inside X and is compact. The remaining cover sets cover K, so finitely many of them suffice. Together with U they form a finite subcover of X*.
This proof reveals why neighbourhoods of infinity are defined by compact complements. One chosen neighbourhood absorbs everything except a compact remainder, and compactness handles that remainder finitely.
The embedded copy of X is open in X*. It is also dense when X is noncompact: every neighbourhood of ∞ meets X. Removing ∞ recovers the original space and topology exactly.
CHAPTER 3 OF 12 · WORK CONCRETE EXAMPLES
3. Work the real line into a circle
For X=R, compact subsets are closed and bounded. Typical neighbourhoods of ∞ contain {∞} together with the two tails (−∞,−M)∪(M,∞). Both directions along the line approach the same added point.
Stereographic projection identifies R with a circle minus one point. Restoring that missing point produces S¹, so the one-point compactification of R is homeomorphic to the circle.
| Feature | Real line R | Compactification R* |
|---|---|---|
| Points | All real numbers | Real numbers plus ∞ |
| Neighbourhood near infinity | Not available | Outside a compact interval, plus ∞ |
| Compact? | No | Yes |
| Familiar model | Open curve | Circle S¹ |
Likewise, the one-point compactification of Rⁿ is homeomorphic to Sⁿ.
CHAPTER 4 OF 12 · WORK CONCRETE EXAMPLES
4. Changed context: a half-line closes as an interval
Take X=[0,∞). The map h(x)=x/(1+x) sends X homeomorphically onto [0,1). Sending ∞ to 1 extends h to a homeomorphism X*→[0,1]. The half-line has one open end, so the added point becomes an endpoint rather than joining two tails into a circle.
| Original space | Escape pattern | One-point compactification |
|---|---|---|
| R | Two tails, identified at ∞ | S¹ |
| [0,∞) | One unbounded end | [0,1] |
| R² | Every planar direction escapes | S² |
| Infinite discrete N | Leave every finite set | Convergent sequence with its limit |
For N with the discrete topology, compact sets are finite, so neighbourhoods of ∞ are cofinite.
CHAPTER 5 OF 12 · WORK CONCRETE EXAMPLES
5. Local compactness separates ordinary points from infinity
To separate x∈X from ∞, choose a neighbourhood V of x whose closure K is compact. Then V and {∞}∪(X\K) are disjoint neighbourhoods. This is the point where local compactness enters the Hausdorff proof.
If X is Hausdorff but not locally compact at x, the same set construction can still be compact as a topological space, yet x and ∞ may fail to have disjoint neighbourhoods. Calling it a compactification without naming the separation quality hides the defect.
In the standard theorem, a noncompact locally compact Hausdorff space has a Hausdorff one-point compactification, unique up to a homeomorphism that fixes X.
CHAPTER 6 OF 12 · USE LOCAL COMPACTNESS
6. Compact subsets control convergence to infinity
A net xα in X converges to ∞ in X* exactly when it eventually lies outside every compact subset of X. In first-countable examples, sequences may suffice. Thus n→∞ in N* and x→±∞ along R both converge to the same compactifying point.
But sequences do not capture all topological convergence in arbitrary spaces. Even in familiar spaces, “the coordinate grows” is only a special case of leaving compact sets.
A sequence that oscillates between n and −n in R still converges to ∞ in R*, because it eventually leaves every bounded closed interval. The compactification forgets which tail was used.
CHAPTER 7 OF 12 · USE LOCAL COMPACTNESS
7. The construction can merge distinct ends
The line has two intuitive ends, but one-point compactification identifies them. A cylinder may have two ends; a plane has one; a tree can have many. End compactifications preserve more directional information by adding more than one boundary point.
One-point compactification is therefore minimal in cardinality, not maximally descriptive. It is ideal when a single point at infinity is the right algebraic or geometric object, as in Rⁿ→Sⁿ. It is less informative when different escape directions matter.
Do not interpret ∞ as a physical location without checking what information the quotient-like identification discards.
CHAPTER 8 OF 12 · USE LOCAL COMPACTNESS
8. Proper maps extend naturally
Let f:X→Y be a proper continuous map between locally compact Hausdorff spaces, meaning preimages of compact sets are compact. Define f*(x)=f(x) on X and f*(∞X)=∞Y. Properness makes f* continuous at infinity: the preimage of a neighbourhood outside compact K⊂Y is outside the compact set f⁻¹(K).
This gives a powerful transfer rule. Questions about proper maps between noncompact spaces can be converted into based continuous maps between their one-point compactifications.
The converse needs the correct hypotheses and definition of properness. Do not extend an arbitrary continuous map by sending infinity to infinity; a constant map on R fails the required escape behaviour.
CHAPTER 9 OF 12 · TEST MAPS AND LIMITS
9. Functions may or may not have a limit at infinity
A continuous f:X→Z into a compact Hausdorff space extends continuously to X* with value L at ∞ precisely when f approaches L outside compact subsets. On R, f(x)=1/(1+x²) extends with value 0. The function sin x does not have a single limit as both tails escape, so it cannot extend continuously to the one-point compactification with one value at ∞.
This example exposes the identification again: a function must agree asymptotically along every way of leaving compact sets. Separate boundary points would permit different limiting data.
Extension claims should state the target space and the meaning of convergence at infinity.
CHAPTER 10 OF 12 · TEST MAPS AND LIMITS
10. Common misconceptions and limits
If X is already compact Hausdorff and the same recipe is applied, ∞ becomes isolated because X itself is compact; X is not dense in the enlarged disjoint union. That is not a compactification in the usual dense-embedding sense.
The added point is not an order maximum. Compactness does not imply metrisability; X* is metrisable under additional conditions, such as suitable countability in common locally compact Hausdorff settings. Connectedness also requires care: one-point compactification of a disconnected space can become connected or remain disconnected depending on its structure.
Finally, this construction differs from the cofinite topology on all of X*. Only neighbourhoods containing ∞ are specified by compact complements; ordinary open sets retain X’s original topology.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions readers should ask
Why must neighbourhoods of infinity have compact complements?
That rule makes one neighbourhood leave a compact remainder, allowing every open cover to reduce to finitely many sets.
Is the one-point compactification always Hausdorff?
Not without the usual local compactness and Hausdorff hypotheses on X.
Why does R become a circle rather than a closed interval?
Both tails approach the same added point. A half-line has one open end and compactifies to an interval.
Does every continuous map extend over infinity?
No. Proper maps extend by ∞→∞, and scalar functions need one consistent limit outside compact sets.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Continue through the verified topology ecosystem
Begin with the Topology root, the Gold Standard of Education and the How X Works hub. Build the compactness argument with Compactness and the Finite Intersection Property and audit the separation hypothesis through Compact Sets, Closedness and the Missing Hausdorff Hypothesis.
For related mapping arguments, read Closed Projections and the Compact Factor That Keeps Limits from Escaping and A Continuous Bijection with a Discontinuous Inverse. Complete Metrisability, G-Delta Subspaces and a Missing Metric separates compactification from metric completeness, while Cofinite Spaces, Compactness and Non-Unique Limits shows why compactness without Hausdorffness behaves differently.
Authoritative technical reading: the University of Chicago’s Basic Topology notes define one-point compactification through compact complements and explain the local-compactness requirement for Hausdorffness. Munkres’ Topology chapter on local compactness, hosted by the University of California, Riverside, develops the existence and uniqueness theorem. Use the hypotheses explicitly rather than treating “add infinity” as a purely set-theoretic step.
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