Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space formula_1. 
It states: 
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.
Notes.
The conclusion of the theorem can equivalently be formulated as: " is an open map".
Normally, to check that is a homeomorphism, one would have to verify that both and its inverse function are continuous; 
the theorem says that if the domain is an "open" subset of formula_1 and the image is also in formula_6 then continuity of is automatic. 
Furthermore, the theorem says that if two subsets "U" and of formula_1 are homeomorphic, and is open, then must be open as well. 
(Note that V is open as a subset of formula_6 and not just in the subspace topology. 
Openness of V in the subspace topology is automatic.) 
Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space.
[[File:A map which is not a homeomorphism onto its image.png|thumb|alt=Not a homeomorphism onto its image|A map which is not a homeomorphism onto its image: 
[[Category:Algebraic topology]]
[[Category:Homeomorphisms]]
[[Category:Theorems in topology]]