##  [Invariance of Domain](/invariance-domain-0) 

 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.

 

 

 

 

 

 





## Principle

Principle

Local Euclidean structure plus injectivity and continuity preserve openness and local homeomorphism properties: embeddings cannot collapse dimension or create boundary points inside the image.

 

 

 

 

 





## Demonstration

Demonstration

If U is an open ball in R^n and f:U→R^n is continuous and injective, then f(U) is open in R^n and f:U→f(U) is a homeomorphism; this prevents, for example, a continuous injective map from mapping an open n-ball into a lower-dimensional subspace.

 

 

 

 

## Misapplication

Misapplication

Assuming the result for injections between spaces of different dimensions, for non-injective continuous maps, or in infinite-dimensional topological vector spaces without verifying extra hypotheses.

 

 

 

 

 





## Consequence

Consequence

Implies invariance of domain dimension-wise and forbids pathological embeddings of Euclidean domains into lower-dimensional Euclidean spaces; it underpins many classification results in topology and manifold theory.

 

 

 

 

## Reversal

Reversal

Dropping injectivity permits images of open sets to fail to be open (they may fold or self-overlap); changing domain or codomain to non-Euclidean or infinite-dimensional spaces can reverse openness conclusions.

 

 

 

 

 





## Boundary

Boundary

Holds for continuous injective maps between open subsets of Euclidean n-space; does not apply to general topological spaces, to maps that are not injective, or to embeddings into spaces lacking local Euclidean structure without further hypotheses.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Related to invariance of dimension and to embedding theorems; tension arises when comparing local topological behavior (invariance of domain) with global manifold embeddings or with results that depend on smoothness or differentiability rather than mere continuity.

 

 

 

 

 





## Synthesis

Synthesis

Invariance of Domain asserts that continuous injective maps between Euclidean domains preserve openness and local homeomorphism structure, formalizing that topological embeddings of n-dimensional Euclidean pieces cannot hide or reduce their dimension.