Inverse Limit - Examples

Examples

• The ring of p-adic integers is the inverse limit of the rings Z/pnZ (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". The natural topology on the p-adic integers is the same as the one described here.
• The ring of formal power series over a commutative ring R can be thought of as the inverse limit of the rings, indexed by the natural numbers as usually ordered, with the morphisms from to 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, f_ij) have a greatest element m. Then the natural projection π_m: X → X_m is an isomorphism.
• Inverse limits in the category of topological spaces are given by placing the initial topology 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).
• Let (I, =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the product.
• Let I consist of three elements i, j, and k with i ≤ j and i ≤ k (not directed). The inverse limit of any corresponding inverse system is the pullback.