Inverse system

Examples

• The ring of p-adic integers is the inverse limit of the rings ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers ${\displaystyle (n_{1},n_{2},\dots )}$ such that each element of the sequence "projects" down to the previous ones, namely, that ${\displaystyle n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}}$ whenever ${\displaystyle i<j.}$ The natural topology on the p-adic integers is the one implied here, namely the product topology with cylinder sets as the open sets.
• The p-adic solenoid is the inverse limit of the topological groups ${\displaystyle \mathbb {R} /p^{n}\mathbb {Z} }$ 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 ${\displaystyle (x_{1},x_{2},\dots )}$ such that each element of the sequence "projects" down to the previous ones, namely, that ${\displaystyle x_{i}\equiv x_{j}{\mbox{ mod }}p^{i}}$ whenever ${\displaystyle i<j.}$ Its elements are exactly of form ${\displaystyle n+r}$, where ${\displaystyle n}$ is a p-adic integer, and ${\displaystyle r\in [0,1)}$ is the "remainder".
• The ring ${\displaystyle \textstyle R[[t]]}$ of formal power series over a commutative ring R can be thought of as the inverse limit of the rings ${\displaystyle \textstyle R[t]/t^{n}R[t]}$, indexed by the natural numbers as usually ordered, with the morphisms from ${\displaystyle \textstyle R[t]/t^{n+j}R[t]}$ to ${\displaystyle \textstyle R[t]/t^{n}R[t]}$ given by the natural projection.
• Pro-finite groups are defined as inverse limits of (discrete) finite groups.
• Let the index set I of an inverse system (X_i, ${\displaystyle f_{ij}}$) have a greatest elementm. Then the natural projection π_m: X → X_m 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 (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 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, the limit space is totally disconnected. This is one way of realizing the p-adic numbers and the Cantor set (as infinite strings).