# Invariance of domain

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

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

The theorem and its proof are due to L. E. J. Brouwer, published in 1912.[1] The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

## Notes

The conclusion of the theorem can equivalently be formulated as: "f is an open map".

Normally, to check that f is a homeomorphism, one would have to verify that both f and its inverse function f −1 are continuous; the theorem says that if the domain is an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and the image is also in ${\displaystyle \mathbb {R} ^{n},}$ then continuity of f −1 is automatic. Furthermore, the theorem says that if two subsets U and V of ${\displaystyle \mathbb {R} ^{n}}$ are homeomorphic, and U is open, then V must be open as well. (Note that V is open as a subset of ${\displaystyle \mathbb {R} ^{n},}$ 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.

g : (−1, 1) → ${\displaystyle \mathbb {R} ^{2}}$ with g(t) = (t2 − 1, t3t)

It is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f: (0,1)${\displaystyle \mathbb {R} ^{2}}$ defined by f(t) = (t, 0). This map is injective and continuous, the domain is an open subset of ${\displaystyle \mathbb {R} }$, but the image is not open in ${\displaystyle \mathbb {R} ^{2}.}$ A more extreme example is the map g: (−1.1, 1) → ${\displaystyle \mathbb {R} ^{2}}$ defined by g(t) = (t 2 − 1, t 3 − t) because here g is injective and continuous but does not even yield a homeomorphism onto its image.

The theorem is also not generally true in infinite dimensions. Consider for instance the Banach space l of all bounded real sequences. Define f : ll as the shift f(x1, x2, ...) = (0, x1, x2, ...). Then f is injective and continuous, the domain is open in l, but the image is not.

## Consequences

An important consequence of the domain invariance theorem is that ${\displaystyle \mathbb {R} ^{n}}$ cannot be homeomorphic to ${\displaystyle \mathbb {R} ^{m}}$ if mn. Indeed, no non-empty open subset of ${\displaystyle \mathbb {R} ^{n}}$ can be homeomorphic to any open subset of ${\displaystyle \mathbb {R} ^{m}}$ in this case.

## Generalizations

The domain invariance theorem may be generalized to manifolds: if M and N are topological n-manifolds without boundary and f : MN is a continuous map which is locally one-to-one (meaning that every point in M has a neighborhood such that f restricted to this neighborhood is injective), then f is an open map (meaning that f(U) is open in N whenever U is an open subset of M) and a local homeomorphism.

There are also generalizations to certain types of continuous maps from a Banach space to itself.[2]

## References

1. ^ Brouwer L.E.J. Beweis der Invarianz des n-dimensionalen Gebiets, Mathematische Annalen 71 (1912), pages 305–315; see also 72 (1912), pages 55–56
2. ^ Leray J. Topologie des espaces abstraits de M. Banach. C. R. Acad. Sci. Paris, 200 (1935) pages 1083–1093