Inverse limit: Difference between revisions
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{{Short description|Construction in category theory}} | {{Short description|Construction in category theory}} | ||
In [[mathematics]], | In [[mathematics]], an '''inverse limit''' (also called a '''projective limit''') is a construction that allows one to "glue together" several related [[mathematical object|objects]], the precise gluing process being specified by [[morphisms]] between the objects. Inverse limits can be defined in any [[category (mathematics)|category]], although their existence depends on the category that is considered. They are a special case of the concept of a [[Limit (category theory)|limit]] in category theory. | ||
By working in the [[dual category]], | By working in the [[dual category]]—that is, by reversing the arrows—an inverse limit becomes a [[direct limit]] or ''inductive limit'', and a ''limit'' becomes a [[colimit]]. | ||
== Formal definition == | == Formal definition == | ||
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* Let the index set ''I'' of an inverse system (''X''<sub>''i''</sub>, <math>f_{ij}</math>) have a [[greatest element]] ''m''. Then the natural projection {{pi}}<sub>''m''</sub>: ''X'' → ''X''<sub>''m''</sub> is an isomorphism. | * Let the index set ''I'' of an inverse system (''X''<sub>''i''</sub>, <math>f_{ij}</math>) have a [[greatest element]] ''m''. Then the natural projection {{pi}}<sub>''m''</sub>: ''X'' → ''X''<sub>''m''</sub> is an isomorphism. | ||
* In the [[category of sets]], every inverse system has an inverse limit, which can be constructed in an elementary manner as a subset of the product of the sets forming the inverse system. The inverse limit of any inverse system of non-empty finite sets is non-empty. This is a generalization of [[Kőnig's lemma]] in graph theory and may be proved with [[Tychonoff's theorem]], viewing the finite sets as compact discrete spaces, and then applying the [[finite intersection property]] characterization of compactness. | * In the [[category of sets]], every inverse system has an inverse limit, which can be constructed in an elementary manner as a subset of the product of the sets forming the inverse system. The inverse limit of any inverse system of non-empty finite sets is non-empty. This is a generalization of [[Kőnig's lemma]] in graph theory and may be proved with [[Tychonoff's theorem]], viewing the finite sets as compact discrete spaces, and then applying the [[finite intersection property]] characterization of compactness. | ||
* In the [[category of topological spaces]], every inverse system has an inverse limit. It is constructed by placing the [[initial topology]] on the underlying set-theoretic inverse limit. This is known as the '''limit topology'''. | * In the [[category of topological spaces]], every inverse system has an inverse limit. It is constructed by placing the [[initial topology]] (with respect to the projection maps into the constituent spaces of the inverse system) on the underlying set-theoretic inverse limit. This is known as the '''limit topology'''. | ||
** The set of infinite [[String (computer science)|strings]] is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are [[discrete topology|discrete]], the limit space is [[totally disconnected]]. This is one way of realizing the [[p-adic|''p''-adic numbers]] and the [[Cantor set]] (as infinite strings). | ** The set of infinite [[String (computer science)|strings]] is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are [[discrete topology|discrete]], the limit space is [[totally disconnected]]. This is one way of realizing the [[p-adic|''p''-adic numbers]] and the [[Cantor set]] (as infinite strings). | ||