Alexandroffkompaktifikasjon

Alexandroff compactification, commonly called the one-point compactification, is a construction in topology that turns a non-compact locally compact space into a compact space by adjoining a single point at infinity. Named after Pavel Alexandroff, it provides a simple but powerful way to study non-compact spaces through compact ones.

Construction and definition. Let X be a locally compact Hausdorff space that is not compact. Form the space X* = X ∪ {∞}. The open sets of X* consist of all open subsets of X, together with sets of the form (X \ K) ∪ {∞}, where K is a compact subset of X. Equivalently, neighborhoods of ∞ are exactly the complements in X* of compact subsets of X. With this topology, X embeds as a dense open subset of X*, and X* \ {∞} is homeomorphic to X. If X is non-compact, X* is a compact Hausdorff space.

Properties and examples. The construction gives a compactification of X that is unique up to a homeomorphism; any two one-point compactifications of the same X are homeomorphic. The one-point compactification preserves many topological properties: X is connected if and only if X* is connected, and X is locally compact and non-compact ensures X* is compact. A standard example is the real line R, whose one-point compactification is homeomorphic to the circle S^1. The point ∞ serves as an idealized “point at infinity” that closes off X.

Applications and context. The Alexandroff compactification is the simplest nontrivial compactification and is used to extend results from compact spaces to certain non-compact settings, to study convergence and boundary behavior, and to provide a compact framework for various analytical and topological arguments.