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 ...
, a
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 poin ...
is called simply connected (or 1-connected, or 1-simply connected) if it is
path-connected and every
path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The
fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.
Definition and equivalent formulations
A
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 poin ...
is called if it is path-connected and any
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
in
defined by
can be contracted to a point: there exists a continuous map
such that
restricted to
is
Here,
and
denotes the
unit circle and closed
unit disk in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
respectively.
An equivalent formulation is this:
is simply connected if and only if it is path-connected, and whenever
and
are two paths (that is, continuous maps) with the same start and endpoint (
and
), then
can be continuously deformed into
while keeping both endpoints fixed. Explicitly, there exists a
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
such that
and
A topological space
is simply connected if and only if
is path-connected and the
fundamental group of
at each point is trivial, i.e. consists only of the
identity element. Similarly,
is simply connected if and only if for all points
the set of
morphisms
in the
fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a ...
of
has only one element.
In
complex analysis: an open subset
is simply connected if and only if both
and its complement in the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes a nice example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. It might also be worth pointing out that a relaxation of the requirement that
be connected leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has a connected extended complement exactly when each of its connected components are simply connected.
Informal discussion
Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are
connected but not simply connected are called non-simply connected or multiply connected.
The definition rules out only
handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of dimension, is called
contractibility.
Examples
* The
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
is simply connected, but
minus the origin
is not. If
then both
and
minus the origin are simply connected.
* Analogously: the
''n''-dimensional sphere is simply connected if and only if
* Every
convex subset of
is simply connected.
* A
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
, the (elliptic)
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an ...
, the
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and A ...
, the
projective plane and the
Klein bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a ...
are not simply connected.
* Every
topological vector space is simply connected; this includes
Banach spaces and
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s.
* For
the
special orthogonal group is not simply connected and the
special unitary group is simply connected.
* The one-point compactification of
is not simply connected (even though
is simply connected).
* The
long line is simply connected, but its compactification, the extended long line
is not (since it is not even path connected).
Properties
A surface (two-dimensional topological
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 ...
) is simply connected if and only if it is connected and its
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
(the number of of the surface) is 0.
A universal cover of any (suitable) space
is a simply connected space which maps to
via a
covering map.
If
and
are
homotopy equivalent and
is simply connected, then so is
The image of a simply connected set under a continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is
which is not simply connected.
The notion of simple connectedness is important in
complex analysis because of the following facts:
* The
Cauchy's integral theorem states that if
is a simply connected open subset of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
and
is a
holomorphic function, then
has an
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
on
and the value of every
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
in
with integrand
depends only on the end points
and
of the path, and can be computed as
The integral thus does not depend on the particular path connecting
and
* The
Riemann mapping theorem states that any non-empty open simply connected subset of
(except for
itself) is
conformally equivalent to the
unit disk.
The notion of simple connectedness is also a crucial condition in the
Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Originally conjectured ...
.
See also
*
*
*
*
*
References
*
*
*
*
*{{cite book , last=Joshi , first=Kapli , title=Introduction to General Topology , date=August 1983 , publisher=New Age Publishers , isbn=0-85226-444-5
Algebraic topology
Properties of topological spaces
de:Zusammenhängender Raum#Einfach zusammenhängend