A geometrical object is called simply connected if it consists of one piece and doesn't have any circle-shaped "holes" or "handles". Higher-dimensional holes are allowed. For instance, a doughnut (with hole) is not simply connected, but a three-dimensional ball (even a hollow one) is. A circle is not simply connected but a disk and a line are. The opposite is non-simply connected or, in a somewhat old-fashioned term, multiply connected.

Informally, suppose someone hands you an object made out of a strong, inflexible material, one that won't bend or break under any condition. Shake the object and turn it every direction you can think of. If anything falls off, rattles, spins, or otherwise moves separately of the object, it's not a "simple" object. Formally, such a simple object is called a connected space, but for our informal definition, we can just think of a simple object as being an object that's all one piece. Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...

Once you have a simple object, take a piece of string and insert one end into the object at any point. Let that end of the string follow any path, leaving behind string everywhere it goes, and then emerge at the same spot it went in, so that you have a loop going through the object.

Now hold on to both ends of the string and maneuver the string inside the object until you are able to pull the loop out through the hole. You may need to feed in some extra string, but that's not a problem. If you can find any path inside the object that makes it impossible to get the loop of string out, the object is not simply connected. If no path from any point of entry gets the loop caught in the object, then it is simply connected.

Notice how this definition does not rule out higher-dimensional holes. For example, while a hollow ball has a 2-dimensional hole in its middle, any loop you tie around the ball you can shrink to a point. The stronger condition, that the object have no holes of any dimension, is called contractibility. This is a glossary of some terms used in the branch of mathematics known as topology. ...

In algebraic topology this idea is made into a formal tool. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...

Naming

• In topology, a connection is often referred to as a handle

This is probably a reference to the way that a (singly-connected) beaker can be topologically turned into a (doubly-connected) teacup by the addition of a handle.

• In theoretical physics, an additional connection is known as a wormhole.

Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Theoretical physics is physics that employs mathematical models and abstractions rather than experimental processes. ... 2D analogy to a wormhole. ...

Formal definition and equivalent formulations

A topological space X is called simply connected if it is path-connected and any continuous map f : S¹ -> X (where S¹ denotes the unit circle in Euclidean 2-space) can be contracted to a point in the following sense: there exists a continuous map F : D² -> X (where D² denotes the unit disk in Euclidean 2-space) such that F restricted to S¹ is f. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... Illustration of a unit circle. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... A disc of unit radius on a plane is called a unit disc. ...

An equivalent formulation is this: X is simply connected if and only if it is path connected, and whenever p : [0,1] → X and q : [0,1] → X are two paths (i.e.: continuous maps) with the same start and endpoint (p(0) = q(0) and p(1) = q(1)), then p and q are homotopic relative {0,1}. Intuitively, this means that p can be "continuously deformed" to get q while keeping the endpoints fixed. Hence the term simply connected: for any two given points in X, there is one and "essentially" only one path connecting them. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

A third way to express the same: X is simply connected if and only if X is path-connected and the fundamental group of X is trivial, i.e. consists only of the identity element. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

Yet another formulation is often used in complex analysis: an open subset X of C is simply connected if and only if both X and its complement in the Riemann sphere are connected. Complex analysis is the branch of mathematics investigating functions of complex numbers. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...

Examples

• The Euclidean plane R² is simply connected, but R² minus the origin (0,0) is not. If n>2, then both Rⁿ and Rⁿ minus the origin are simply connected.
• Analogously: the n-dimensional sphere Sⁿ is simply connected if and only if n≥2.
• A torus, the Möbius band and the Klein bottle are not simply connected.
• Every topological vector space is simply connected; this includes Banach spaces and Hilbert spaces.
• The special orthogonal group SO(n,R) is not simply connected for n≥2; the special unitary group SU(n) is simply connected.
• The long line L is simply connected, but its compactification, the extended long line L* is not (since it is not even path connected).

In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... A sphere is a perfectly symmetrical geometrical object. ... A torus. ... A Möbius strip made with a piece of paper and tape. ... The Klein bottle immersed in three-dimensional space. ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the special unitary group of degree is the group of by unitary matrices with determinant and entries from the field of complex numbers, with the group operation that of matrix multiplication. ... In topology, the long line is a topological space analogous to the real line, but much longer. ...

Properties

A surface (two-dimensional topological manifold) is simply connected if and only if it is connected and its genus is 0. Intuitively, the genus is the number of "holes" or "handles" of the surface. A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...

If a space X is not simply connected, one can often rectify this defect by using its universal cover, a simply connected space which maps to X in a particularly nice way. In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...

If X and Y are homotopy equivalent and X is simply connected, then so is Y. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

Note that the image of a simply connected set under a continuous function need not to be simply connected. Take for example the complex plane under the exponential map, the image is C - {0}, which clearly is not simply connected.

The notion of simply connectedness is important in complex analysis because of the following facts: Complex analysis is the branch of mathematics investigating functions of complex numbers. ...

• If U is a simply connected open subset of the complex plane C, and f : U -> C is a holomorphic function, then f has an antiderivative F on U, and the value of every path integral in U with integrand f depends only on the end points u and v of the path, and can be computed as F(v) - F(v). The integral thus does not depend on the particular path connecting u and v.
• The Riemann mapping theorem states that any non-empty open simply connected subset of C (except for C itself) can be conformally and bijectively mapped to the unit disk.

In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. ... The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the... In mathematics, a conformal map is a function which preserves angles. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

See also

• connected space
• unicoherent