Invariance of domain
is a theorem in topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
about homeomorphic subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. It states:
- If U is an open subset
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
of Rn and f : U → Rn is an injective continuous map, then V = f(U) is open and f is a homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
between U and V.
The theorem and its proof are due to L.E.J. Brouwer, published in 1912. The proof uses tools of algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, notably the Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
An important consequence of the domain invariance theorem is that Rn
cannot be homeomorphic to Rm
. Indeed, no non-empty open subset of Rn
can be homeomorphic to any open subset of Rm
in this case. (Proof: If m
, then we can view Rm
as a subspace of Rn
, and the non-empty open subsets of Rm
are not open when considered as subsets of Rn
. We apply the theorem in the space Rn
The domain invariance theorem may be generalized to manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s: if M
are topological n
-manifolds without boundary and f
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
) is open in N
is an open subset of M
There are also generalizations to certain types of continuous maps from a Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
- Open mapping theorem
Open mapping theorem may refer to:* Open mapping theorem or Banach–Schauder theorem states that a surjective continuous linear transformation of a Banach space X onto a Banach space Y is an open mapping...
for other conditions that ensure that a given continuous map is open.