Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. It states:

If ${\displaystyle U}$ is an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle f:U\rightarrow \mathbb {R} ^{n}}$ is an injective continuous map, then ${\displaystyle V:=f(U)}$ is open in ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle f}$ is a homeomorphism between ${\displaystyle U}$ and ${\displaystyle V}$.

The theorem and its proof are due to L. E. J. Brouwer, published in 1912. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.