One-point compactification

Core idea

Let $X$ be a noncompact, locally compact Hausdorff space. Its one-point compactification is the set

X^+=X\sqcup\{\infty\}

with the original open subsets of $X$ , together with neighborhoods of $\infty$ of the form

\{\infty\}\cup(X\setminus K),

where $K\subseteq X$ is compact. This topology makes $X^+$ a compact Hausdorff space containing $X$ as an open dense subspace.

Characterization

Up to a homeomorphism fixing $X$ , $X^+$ is the unique compact Hausdorff space obtained from $X$ by adjoining exactly one point. Local compactness supplies enough neighborhoods to separate $\infty$ from points of $X$ .

Examples

The one-point compactification of $\mathbb R$ is a circle.
The one-point compactification of $\mathbb R^n$ is the sphere $S^n$ .
The one-point compactification of $\mathbb C$ is the Riemann sphere $\mathbb C\cup\{\infty\}$ .

Convention

Some authors also apply the construction to compact $X$ , in which case the adjoined point is isolated. Restricting to noncompact $X$ gives the dense-embedding characterization above.

References

John M. Lee, Introduction to Topological Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: compactifications and locally compact Hausdorff spaces.