##  [Invariance of Domain](/index.php/invariance-domain) 

  ##  [Invariance of Domain](https://mathlogic.quantumdictionary.io/invariance-domain-0) 

  

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

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The topological theorem stating that an injective continuous map between open subsets of Euclidean n-space is an open embedding; in particular, the image of an open set in R^n under such a map is open and the map is a homeomorphism onto its image.

 

 

 

 

 





 

 



 ##  [Invariance of Domain](https://puremath.quantumdictionary.io/invariance-domain-1) 

  

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

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A topological principle (Brouwer's Invariance of Domain) that a continuous injective map between n-dimensional Euclidean manifolds (or between open subsets of R^n) is an open embedding; equivalently, a continuous injective map from an open set of R^n into R^n has open image. A key consequence is that Euclidean spaces R^m and R^n are not homeomorphic when m ≠ n.