##  [Completion](/completion) 

  ##  [Completion](https://mathlogic.quantumdictionary.io/completion-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The procedure of adjoining limits of Cauchy nets, sequences, or more general Cauchy filters to a metric, uniform, or algebraic object to obtain a complete object; common instances include Cauchy completion of metric spaces and I-adic completion of rings.

 

 

 

 

 





 

 



 ##  [Completion](https://puremath.quantumdictionary.io/completion-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The process of adjoining limits of Cauchy sequences (or inverse‑limit points) with respect to a topology or filtration to obtain a complete object in which prescribed Cauchy nets converge.

 

 

 

 

 





 

 



 ##  [Completion](https://algebra.quantumdictionary.io/completion-2) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The process of adjoining limits of Cauchy-like sequences (or inverse-limit elements determined by a filtration or topology) to an algebraic object so that the resulting object is complete with respect to the given topology or filtration.