Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space

R

n

{\displaystyle \mathbb {R} ^{n}}

.

It states:

If

U

{\displaystyle U}

is an open subset of

R

n

{\displaystyle \mathbb {R} ^{n}}

and

f

:

U

→

R

n

{\displaystyle f:U\rightarrow \mathbb {R} ^{n}}

is an injective continuous map, then

V

:=

f

(

U

)

{\displaystyle V:=f(U)}

is open in

R

n

{\displaystyle \mathbb {R} ^{n}}

and

f

{\displaystyle f}

is a homeomorphism between

U

{\displaystyle U}

and

V

{\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.

Notes

The conclusion of the theorem can equivalently be formulated as: "

f

{\displaystyle f}

is an open map".

Normally, to check that

f

{\displaystyle f}

is a homeomorphism, one would have to verify that both

f

{\displaystyle f}

and its inverse function

f

−

1

{\displaystyle f^{-1}}

are continuous;

the theorem says that if the domain is an open subset of

R

n

{\displaystyle \mathbb {R} ^{n}}

and the image is also in

R

n

,

{\displaystyle \mathbb {R} ^{n},}

then continuity of

f

−

1

{\displaystyle f^{-1}}

is automatic.

Furthermore, the theorem says that if two subsets

U

{\displaystyle U}

and

V

{\displaystyle V}

of

R

n

{\displaystyle \mathbb {R} ^{n}}

are homeomorphic, and

U

{\displaystyle U}

is open, then

V

{\displaystyle V}

must be open as well.

(Note that

V

{\displaystyle V}

is open as a subset of

R

n

,

{\displaystyle \mathbb {R} ^{n},}

and not just in the subspace topology.

Openness of

V

{\displaystyle 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.

It is of crucial importance that both domain and image of

f

{\displaystyle f}

are contained in Euclidean space of the same dimension.