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 po ...
is called simply connected (or 1-connected, or 1-simply connected) if it is
path-connected and every
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
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 po ...
is called if it is path-connected and any
loop 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
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
and closed
unit disk in the
Euclidean plane 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 defor ...
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
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
. 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 to ...
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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
but not simply connected are called non-simply connected or multiply connected.
The definition rules out only
handle
A handle is a part of, or attachment to, an object that allows it to be grasped and manipulated by hand. The design of each type of handle involves substantial ergonomic issues, even where these are dealt with intuitively or by following tr ...
-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 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
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
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 not tou ...
, 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 infin ...
, the
Möbius strip, the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
and the
Klein bottle are not simply connected.
* Every
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is simply connected; this includes
Banach spaces and
Hilbert spaces.
* For
the
special orthogonal group is not simply connected and the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
is simply connected.
* The one-point compactification of
is not simply connected (even though
is simply connected).
* The
long line Long line or longline may refer to:
*'' Long Line'', an album by Peter Wolf
* Long line (topology), or Alexandroff line, a topological space
*Long line (telecommunications), a transmission line in a long-distance communications network
*Longline fi ...
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) 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 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 ...
.
If
and
are
homotopy equivalent
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 deforma ...
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
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
states that if
is a simply connected open subset of the
complex plane 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 symbolicall ...
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, al ...
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
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphi ...
states that any non-empty open simply connected subset of
(except for
itself) is
conformally equivalent
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal map, conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space high ...
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