Invariance of domain is a theorem in
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
about
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
.
It states:
:If
is an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of
and
is an
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
, then
is open in
and
is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
between
and
.
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 that 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 ...
, notably the
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simple ...
.
Notes
The conclusion of the theorem can equivalently be formulated as: "
is an
open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
".
Normally, to check that
is a homeomorphism, one would have to verify that both
and its
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X ...
are continuous;
the theorem says that if the domain is an subset of
and the image is also in
then continuity of
is automatic.
Furthermore, the theorem says that if two subsets
and
of
are homeomorphic, and
is open, then
must be open as well.
(Note that
is open as a subset of
and not just in the subspace topology.
Openness of
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
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
and
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of
are contained in Euclidean space .
Consider for instance the map
defined by
This map is injective and continuous, the domain is an open subset of
, but the image is not open in
A more extreme example is the map
defined by
because here
is injective and continuous but does not even yield a homeomorphism onto its image.
The theorem is also not generally true in infinitely many dimensions. Consider for instance the
Banach lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
of all bounded real
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
s.
Define
as the shift
Then
is injective and continuous, the domain is open in
, but the image is not.
Consequences
An important consequence of the domain invariance theorem is that
cannot be homeomorphic to
if
Indeed, no non-empty open subset of
can be homeomorphic to any open subset of
in this case.
Generalizations
The domain invariance theorem may be generalized to
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s: if
and
are topological -manifolds without boundary and
is a continuous map which is
locally one-to-one (meaning that every point in
has a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
such that
restricted to this neighborhood is injective), then
is an
open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
(meaning that
is open in
whenever
is an open subset of
) and a
local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an à ...
.
There are also generalizations to certain types of continuous maps from a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
to itself.
[ Topologie des espaces abstraits de M. Banach. ''C. R. Acad. Sci. Paris'', 200 (1935) pages 1083–1093]
See also
* for other conditions that ensure that a given continuous map is open.
Notes
References
*
*
*
*
*
* (see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction)
*
*
*
*
*
*
*
External links
*
{{Topology
Algebraic topology
Theory of continuous functions
Homeomorphisms
Theorems in topology