Local Compactness

Topology · Axiom Academy

Understanding compactness in neighborhoods and the one-point compactification While many important spaces are not compact themselves, they often have a weaker property: every point has a compact neighborhood . This is the essence of local compactness. A topological space is called locally compact if every point has a compact neighborhood. That is, for each , there exists a compact set and an open set such that . The most important examples of locally compact spaces are the Euclidean spaces . For any point in , we can choose a closed ball of radius . Since closed and bounded sets in are compact (Heine-Borel theorem), this closed ball is a compact neighborhood of . ℝ (the real line): Every point has a compact interval around it. ℝ² (the plane): Every point has a compact disk around it. ℝ³ (3D space): Every point has a compact ball around it. ℝⁿ (n-dimensional space): Every point has a compact n-ball around it. Counterexample: When Local Compactness Fails Not every space is locally compact. The most important counterexample comes from infinite-dimensional spaces. The rationals with the subspace topology from are not locally compact . Every neighborhood of a rational number contains irrational numbers in its closure, and no subset of containing a non-empty open set can be compact.

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