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}} .

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

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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Inverse limit by

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The inverse limit (or projective limit) is a concept in topology and abstract algebra that generalizes the notion of taking a limit of sequences or families of objects. It is particularly useful in the study of topological spaces, algebraic structures, and their relationships.

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