The construction that adjoins a single new point ∞ to a noncompact space X and equips X ∪ {∞} with the topology whose open sets are the opens of X together with sets whose complement in X is compact; for locally compact Hausdorff X this produces a compact Hausdorff space in which X embeds as an open dense subspace.

Principle

Make the new point’s neighborhoods be complements of compact subsets of X, producing the smallest topology that compactifies X by a single point while preserving the original opens and continuity of inclusion.

Demonstration

The one‑point compactification of R is homeomorphic to S^1: add ∞ and declare neighborhoods of ∞ to be complements of compact subsets of R so that sequences escaping to ±∞ converge to ∞, yielding a compact space.

Misapplication

Applying the construction to spaces that are not locally compact when one expects a Hausdorff compactification; the resulting space can fail to be Hausdorff or to reflect desired separation properties.

Consequence

When applicable (locally compact Hausdorff noncompact X), the construction yields a compactification minimal in the sense of adding a single point and provides a convenient compact ambient space for extending continuous functions that vanish at infinity.

Reversal

Removing the adjoined point and its neighborhood rule restores noncompactness; alternatively, attempting to compactify by adjoining too many points or altering neighborhood definitions produces different (often larger) compactifications.

Boundary

Primarily intended for noncompact spaces and yields a Hausdorff compactification exactly when X is locally compact and Hausdorff; it excludes constructions that require universality or that extend arbitrary continuous maps uniquely (unlike Stone‑Čech).

Semantic Tension

Tension exists between one‑point compactification and universal compactifications: the one‑point method is economical but not universal — it may not extend all continuous maps to arbitrary compact targets — whereas Stone‑Čech is universal but much larger.

Synthesis

The one‑point compactification adjoins a point whose neighborhoods are complements of compact sets of X to turn 'escape to infinity' into convergence to a single point; for locally compact Hausdorff spaces this yields a compact Hausdorff space embedding X as an open dense subspace and providing a simple compactification.