Here are some examples of inverse limits of sets.
-
The $p$-Adic Integers. The ring of $p$-adic integers $\mathbb {Z}_{p}$ of
is the inverse limit
\[ \mathbb {Z}_{p}\cong \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{n\in \mathbb {N}}\webleft (\mathbb {Z}_{/p^{n}}\webright ); \]see
.
-
Rings of Formal Power Series. The ring $R[\mspace {-3mu}[t]\mspace {-3mu}]$ of formal power series in a variable $t$ is the inverse limit
\[ R[\mspace {-3mu}[t]\mspace {-3mu}]\cong \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{n\in \mathbb {N}}\webleft (R\webleft [t\webright ]/t^{n}R\webleft [t\webright ]\webright ); \]
see
.
-
Profinite Groups. Profinite groups are inverse limits of finite groups; see .
