One Point CompactificationEdit
One Point Compactification, also known as the Alexandroff compactification, is a standard construction in topology that turns a non-compact, locally compact space into a compact space by adjoining a single new point. The resulting space is usually denoted αX, where X is the original space and the new point is often written as ∞. Intuitively, all the ways for sequences or nets to “escape to infinity” in X are redirected to this single point, yielding a compact ambient space in which X sits as a dense open subspace.
This construction plays a central role in the study of compactifications, offering a simple and canonical way to complete spaces without introducing a large remainder. It has concrete geometric interpretations, for example by turning Euclidean space into a sphere, and it interacts nicely with many standard topological properties.
Construction
- Let X be a locally compact, Hausdorff space that is not compact. Form the set αX = X ∪ {∞}, where ∞ is a new point.
- The topology on αX is defined so that:
- The open sets that do not contain ∞ are exactly the open sets of X (viewed as a subset of αX).
- A set U containing ∞ is open in αX if and only if X \ U is compact in X.
- With this topology, αX is a compact space, and X embeds into αX as an open dense subspace via x ↦ x.
- If X is not only locally compact but also Hausdorff, then αX is compact and Hausdorff. Conversely, if X is not locally compact or not Hausdorff, the resulting αX may fail to be compact or to be Hausdorff.
This gives a universal way to “add infinity” to X in a controlled manner, preserving much of the structure of X while ensuring compactness in the simplest possible way.
Properties
- Uniqueness up to homeomorphism: If X is locally compact and Hausdorff, the one-point compactification αX is unique up to a homeomorphism that fixes X. In this sense, it is the canonical single-point way to compactify X.
- Embedding: X is a dense open subspace of αX, and ∞ is the only point added to perform the compactification.
- Neighborhoods of ∞: The neighborhoods of ∞ are precisely the sets of the form {∞} ∪ (X \ K), where K is a compact subset of X.
- Relationship to Hausdorffness: If X is Hausdorff and locally compact, αX is a compact Hausdorff space. If X fails these conditions, αX need not be Hausdorff or compact.
- Limiting behavior: Sequences or nets that would leave every compact subset of X have their limit represented by ∞ in αX.
Examples
- Real Euclidean space: The one-point compactification of ℝ^n is homeomorphic to the n-sphere S^n. Concretely, adding ∞ to ℝ^n and topologizing as above yields a space that is topologically equivalent to the surface of an (n+1)-dimensional ball.
- Complex plane and the Riemann sphere: The one-point compactification of the complex plane ℂ is the Riemann sphere, homeomorphic to S^2. This provides a geometric realization of complex analysis on a compact surface.
- Discrete infinite spaces: If X is an infinite discrete space, the compact subsets are finite, and αX adds ∞ with neighborhoods that are cofinite in X together with ∞. This produces a compact space in which X sits as a dense open subset, with all “points at infinity” identified as the single ∞.
Flexibility and limits
- Existence and scope: The one-point compactification exists and behaves well for spaces that are locally compact and Hausdorff. It is not suitable for spaces that fail these conditions, and other compactifications (such as the Stone–Čech compactification) may be used in those contexts.
- Comparison with other compactifications: αX is a particularly small and simple compactification, adding only one point. Other compactifications, like the Stone–Čech or the Čech–Stone–Čech compactification, can add a large, complicated remainder and are tailored for different purposes in the study of continuous functions and extensions.