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In mathematics, well behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in . The reason to do this was in line with an idea of making topology, more precisely algebraic topology, more geometric. Localization of a space ''X'' is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space ''X'', directly, giving a second space ''Y''. ==Definitions== We let ''A'' be a subring of the rational numbers, and let ''X'' be a simply connected CW complex. Then there is a simply connected CW complex ''Y'' together with a map from ''X'' to ''Y'' such that *''Y'' is ''A''-local; this means that all its homology groups are modules over ''A'' *The map from ''X'' to ''Y'' is universal for (homotopy classes of) maps from ''X'' to ''A''-local CW complexes. This space ''Y'' is unique up to homotopy equivalence, and is called the localization of ''X'' at ''A''. If ''A'' is the localization of Z at a prime ''p'', then the space ''Y'' is called the localization of ''X'' at ''p'' The map from ''X'' to ''Y'' induces isomorphisms from the ''A''-localizations of the homology and homotopy groups of ''X'' to the homology and homotopy groups of ''Y''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Localization of a topological space」の詳細全文を読む スポンサード リンク
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