A topological construction that adjoins a single new point (often written ∞) to a noncompact, locally compact Hausdorff space so that the enlarged space is compact; neighborhoods of the adjoined point are taken to be complements of compact subsets of the original space.

Principle

Attach one point whose neighborhoods are complements of compact sets in the original space; this produces a compact space when the original is locally compact and Hausdorff, and yields the minimal compactification that identifies all ‘directions to infinity’ into one point.

Demonstration

The one-point compactification of R^n is homeomorphic to the n-sphere S^n: stereographic projection exhibits R^n ∪ {∞} ≅ S^n. For a noncompact connected manifold, adjoining one point often yields a compact manifold with a single added point at infinity.

Misapplication

Trying to form a one-point compactification of a space that is not locally compact (for example, an infinite product of nontrivial spaces) and expecting the result to be Hausdorff; in such cases the resulting topology may fail separation axioms or fail to be compact in the intended way.

Consequence

When correctly applied, the construction yields a compact Hausdorff space with a canonical embedding of the original space as a dense open subset and gives a minimal compactification collapsing all ends to a single point.

Reversal

Removing a point from a compact Hausdorff space that is homeomorphic to a one-point compactification recovers a noncompact locally compact Hausdorff space; e.g., S^n \\{p} ≅ R^n.

Boundary

Defined and useful for noncompact, locally compact Hausdorff spaces. For spaces that are already compact, or not locally compact, the construction either produces a trivial isolated point or fails to preserve Hausdorff-ness and so falls outside the intended scope.

Semantic Tension

Competes with multi-point compactifications (e.g., Stone-Čech) and with compactifications that preserve more function-theoretic data; one-point compactification is the minimal 'collapse all infinities' option, while others distinguish different directions to infinity.

Synthesis

One-point compactification is the process of adding a single ideal point whose neighborhoods are complements of compact sets to convert a noncompact, locally compact Hausdorff space into a compact Hausdorff space; it is the simplest compactification that identifies all ends into one point and is especially natural for manifolds and Euclidean spaces.