Editing Inverse limit (section)

== Examples ==
* The ring of [[p-adic number|''p''-adic integers]] is the inverse limit of the rings <math>\mathbb{Z}/p^n\mathbb{Z}</math> (see [[modular arithmetic]]) with the index set being the [[natural number]]s with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers <math>(n_1, n_2, \dots)</math> such that each element of the sequence "projects" down to the previous ones, namely, that <math>n_i\equiv n_j \mbox{ mod } p^{i}</math> whenever <math>i<j.</math> The natural topology on the ''p''-adic integers is the one implied here, namely the [[product topology]] with [[cylinder set]]s as the open sets.
* The [[Solenoid (mathematics)|''p''-adic solenoid]] is the inverse limit of the topological groups <math>\mathbb{R}/p^n\mathbb{Z}</math> with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of real numbers <math>(x_1, x_2, \dots)</math> such that each element of the sequence "projects" down to the previous ones, namely, that <math>x_i\equiv x_j \mbox{ mod } p^{i}</math> whenever <math>i<j.</math> Its elements are exactly of form <math>n + r</math>, where <math>n</math> is a ''p''-adic integer, and <math>r\in [0, 1)</math> is the "remainder".
* The ring <math>\textstyle R[[t]]</math> of [[formal power series]] over a commutative ring ''R'' can be thought of as the inverse limit of the rings <math>\textstyle R[t]/t^nR[t]</math>, indexed by the natural numbers as usually ordered, with the morphisms from <math>\textstyle R[t]/t^{n+j}R[t]</math> to <math>\textstyle R[t]/t^nR[t]</math> given by the natural projection.
* [[Pro-finite group]]s are defined as inverse limits of (discrete) finite groups.
* 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 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'''.
** 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).