Inverse system

An inverse system is a family

(N_{i})

with transition maps

g_{ij} : N_{j} \to N_{i}

for

i \leq j

, satisfying

g_{ii} = 1

and

g_{ik} = g_{ij} g_{jk}

.

Table of contents
- Inverse limit Inverse system
  - Universal property of an inverse limit Inverse limit
  - Nonemptiness theorem for inverse limits of finite sets Inverse limit

Inverse limit ( $lim N_{i}$ )

The inverse limit is the module of compatible families

lim_{i} N_{i} = {(x_{i}) \in \prod_{i} N_{i} : g_{ij} (x_{j}) = x_{i}} .

Universal property of an inverse limit

Maps from a module

X

to

lim_{i} N_{i}

correspond naturally to compatible families of maps

X \to N_{i}

.

Nonemptiness theorem for inverse limits of finite sets

An inverse system of nonempty finite sets over a directed index set has nonempty inverse limit. Give each set the discrete topology: the compatibility conditions are closed subsets of the compact product, and directedness gives them the finite intersection property.

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