Locally compact space

Locally compact space
Name	Locally compact space
Field	Topology
Notable examples	Euclidean space, topological manifolds, Lie groups, discrete spaces

Locally compact space

A locally compact space is a topological concept concerning points that have neighborhoods whose closures are compact; it sits centrally among notions studied in topology, functional analysis, and geometric group theory. The concept has consequences for measure theory, harmonic analysis, and the structure of topological groups, and interacts with classical results associated with Euclid, Hilbert, and Hausdorff. Historically linked to work by Henri Lebesgue, Maurice Fréchet, and Felix Hausdorff, local compactness underpins much of modern analysis on manifolds, Lie groups, and locally compact abelian groups.

Definition and basic properties

A space is locally compact if every point admits a neighborhood with compact closure, a notion formalized in the work of Felix Hausdorff and Maurice Fréchet and used in results by Henri Lebesgue and Andrey Kolmogorov on measure and integration. Elementary properties connect local compactness with separation axioms studied by Georg Cantor and David Hilbert; for example, a locally compact Hausdorff space often satisfies regularity conditions used in theorems of L. E. J. Brouwer and Emil Artin. Local compactness is preserved under classical operations like taking open subspaces and finite disjoint unions, with proofs relying on techniques from Élie Cartan and John von Neumann appearing in functional-analytic contexts. Many standard lemmas and propositions about locally compact spaces are employed in constructions by Hermann Weyl, Jean-Pierre Serre, and Alexander Grothendieck.

Examples

Basic examples include Euclidean spaces studied by René Descartes and Isaac Newton and compact manifolds encountered in the work of Bernhard Riemann and Henri Poincaré, while discrete spaces featured in combinatorial studies by Paul Erdős and Alfréd Rényi are trivially locally compact. Lie groups examined by Sophus Lie and further developed by Élie Cartan provide rich noncompact locally compact examples, as do p-adic numbers used by Kurt Hensel and John Tate in number theory. Nonexamples arise in constructions by Felix Hausdorff and counterexamples exhibited in texts by Kelley and Munkres, including the infinite product of noncompact spaces whose failure traces back to phenomena observed by Stefan Banach and Alfréd Haar.

Equivalent formulations and variations

Several equivalent formulations relate local compactness to properties proved in works by Andrey Kolmogorov and Pavel Alexandrov; for instance, in Hausdorff settings one may require each point to have a neighborhood whose closure is compact or a local base of relatively compact neighborhoods as used by Marshall Stone and John von Neumann. Variations include local compactness at a point and sigma-compact local compactness considered in measure-theoretic expositions by Henri Lebesgue and Norbert Wiener, as well as one-point compactifications introduced by Maurice Fréchet and formalized by Alexandre Grothendieck in categorical contexts. Interplay with paracompactness and metrizability appears in theorems by James Munkres and Mary Ellen Rudin, with criteria influenced by techniques from Riesz and Frigyes Riesz in functional analysis.

Products, subspaces, and quotient spaces

Local compactness behaves predictably for open subspaces and finite products, results appearing in standard treatments by Kelley and Munkres and used in constructions by H. Cartan and Jean Leray. Infinite products may fail to be locally compact, a phenomenon explored by Stefan Banach and exemplified in functional-analytic spaces studied by John von Neumann and Fréchet. Closed subspaces of locally compact Hausdorff spaces remain locally compact under hypotheses often cited in texts by Dieudonné and Bourbaki, while quotient maps can destroy local compactness unless additional conditions from work by Samuel Eilenberg and Norman Steenrod are met. One-point compactification, a classical operation due to Maurice Fréchet and employed by Riesz and Alexandre Grothendieck, produces compact spaces from locally compact Hausdorff spaces.

Relationship with compactness and connectedness

Local compactness refines compactness studied by Kurt Gödel and John von Neumann: every compact space is locally compact, a fact used in proofs by Emil Artin and Henri Lebesgue, but the converse need not hold, a distinction highlighted in examples from Riemannian geometry by Bernhard Riemann and Henri Poincaré. The interaction with connectedness arises in manifold theory developed by André Weil and Hermann Weyl, where locally compact connected manifolds provide the setting for classical theorems by Carl Friedrich Gauss and Élie Cartan. Results about components and local compactness draw on techniques from algebraic topology initiated by Henri Poincaré and expanded by Samuel Eilenberg and Leray.

Local compactness in topological groups and manifolds

Topological groups studied by Haar and André Weil often assume local compactness to ensure the existence of Haar measure, a foundational result of Alfréd Haar and exploited by John von Neumann and Norbert Wiener. Locally compact Lie groups analyzed by Sophus Lie and Élie Cartan underpin representation theory developed by Hermann Weyl and Harish-Chandra, while locally compact manifolds feature in the geometric work of Bernhard Riemann, Henri Poincaré, and Élie Cartan. The structure theory of locally compact groups, including results by George Mackey and John von Neumann, is central to harmonic analysis used by Norbert Wiener and Salomon Bochner.

Applications and significance in analysis and topology

Local compactness is pivotal in establishing the existence of measures and partitions of unity used in integration theories of Henri Lebesgue and Laurent Schwartz, and in enabling functional-analytic constructions by Stefan Banach and Marshall Stone. It underlies the one-point compactification used in algebraic topology by Samuel Eilenberg and Norman Steenrod and the study of duality for locally compact abelian groups due to Lev Pontryagin and Andrey Kolmogorov. Applications range across representation theory of Lie groups, harmonic analysis on p-adic numbers, and index theorems in geometry inspired by Atiyah and Singer.

Category:Topology