In
mathematics, more specifically
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 ...
, a local homeomorphism is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s that, intuitively, preserves local (though not necessarily global) structure.
If
is a local homeomorphism,
is said to be an étale space over
Local homeomorphisms are used in the study of
sheaves. Typical examples of local homeomorphisms are
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
s.
A topological space
is locally homeomorphic to
if every point of
has a neighborhood that is
homeomorphic to an open subset of
For example, a
manifold of dimension
is locally homeomorphic to
If there is a local homeomorphism from
to
then
is locally homeomorphic to
but the converse is not always true.
For example, the two dimensional
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
, being a manifold, is locally homeomorphic to the plane
but there is no local homeomorphism
Formal definition
A function
between two
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s is called a
if for every point
there exists an
open set
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 su ...
containing
such that the
image is open in
and the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
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 isomor ...
(where the respective
subspace topologies are used on
and on
).
Examples and sufficient conditions
Local homeomorphisms versus homeomorphisms
Every homeomorphism is a local homeomorphism. But a local homeomorphism is a homeomorphism if and only if it is
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
.
A local homeomorphism need not be a homeomorphism. For example, the function
defined by
(so that geometrically, this map wraps the
real line around the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
) is a local homeomorphism but not a homeomorphism.
The map
defined by
which wraps the circle around itself
times (that is, has
winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
), is a local homeomorphism for all non-zero
but it is a homeomorphism only when it is
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
(that is, only when
or
).
Generalizing the previous two examples, every
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
is a local homeomorphism; in particular, the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of a space
is a local homeomorphism.
In certain situations the converse is true. For example: if
is a
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
local homeomorphism between two
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s and if
is also
locally compact, then
is a covering map.
Local homeomorphisms and composition of functions
The
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of two local homeomorphisms is a local homeomorphism; explicitly, if
and
are local homeomorphisms then the composition
is also a local homeomorphism.
The restriction of a local homeomorphism to any open subset of the domain will again be a local homomorphism; explicitly, if
is a local homeomorphism then its restriction
to any
open subset of
is also a local homeomorphism.
If
is continuous while both
and
are local homeomorphisms, then
is also a local homeomorphism.
Inclusion maps
If
is any subspace (where as usual,
is equipped with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced by
) then the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota ...
is always a
topological embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
. But it is a local homeomorphism if and only if
is open in
The subset
being open in
is essential for the inclusion map to be a local homeomorphism because the inclusion map of a non-open subset of
yields a local homeomorphism (since it will not be an open map).
The restriction
of a function
to a subset
is equal to its composition with the inclusion map
explicitly,
Since the composition of two local homeomorphisms is a local homeomorphism, if
and
are local homomorphisms then so is
Thus restrictions of local homeomorphisms to open subsets are local homeomorphisms.
Invariance of domain
Invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n.
It states:
:If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorph ...
guarantees that if
is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
injective map
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 ...
from an open subset
of
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 isomor ...
.
Consequently, a continuous map
from an open subset
will be a local homeomorphism if and only if it is a
''locally'' injective map (meaning that every point in
has a
neighborhood such that the restriction of
to
is injective).
Local homeomorphisms in analysis
It is shown in
complex analysis that a complex
analytic function
(where
is an open subset of the
complex plane ) is a local homeomorphism precisely when the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is non-zero for all
The function
on an open disk around
is not a local homeomorphism at
when
In that case
is a point of "
ramification" (intuitively,
sheets come together there).
Using the
inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
one can show that a continuously differentiable function
(where
is an open subset of
) is a local homeomorphism if the derivative
is an invertible linear map (invertible square matrix) for every
(The converse is false, as shown by the local homeomorphism
with
).
An analogous condition can be formulated for maps between
differentiable manifolds.
Local homeomorphisms and fibers
Suppose
is a continuous
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
surjection between two
Hausdorff second-countable spaces where
is a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
and
is a
normal space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
. If every
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
of
is a
discrete subspace of
(which is a necessary condition for
to be a local homeomorphism) then
is a
-valued local homeomorphism on a dense open subset of
To clarify this statement's conclusion, let
be the (unique) largest open subset of
such that
is a local homeomorphism.
[The assumptions that is continuous and open imply that the set is equal to the union of all open subsets of such that the restriction is an ]injective map
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 ...
.
If every
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
of
is a
discrete subspace of
then this open set
is necessarily a
subset of
In particular, if
then
a conclusion that may be false without the assumption that
's fibers are discrete (see this footnote
[Consider the continuous open surjection defined by The set for this map is the empty set; that is, there does not exist any non-empty open subset of for which the restriction is an injective map.] for an example).
One corollary is that every continuous open surjection
between
completely metrizable In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' ind ...
second-countable spaces that has
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
fibers is "almost everywhere" a local homeomorphism (in the topological sense that
is a dense open subset of its domain).
For example, the map
's_domain._
Because_Fundamental_theorem_of_algebra.html" ;"title=", \infty) defined by the polynomial
is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset
's domain.
Because
_(and_thus_a_discrete,_and_even_compact,_subspace),_this_example_generalizes_to_such_polynomials_whenever_the_mapping_induced_by_it_is_an_open_map.