In topology, a geometrical object or space is called simply connected if it is path-connected and every path between two points can be continuously transformed into every other. Informally, an object is simply connected if it consists of one piece and doesn't have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with 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, in a somewhat old-fashioned term, multiply connected. 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. ... 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. ...

Informally, suppose we are considering an object in three dimensions. Think of the object as a strangely shaped aquarium full of water, with rigid sides. Now think of a diver who takes a long piece of string and trails it through the water inside the aquarium, in whatever way he pleases, and then joins the two ends of the string to form a closed loop. Now the loop begins to contract on itself, getting smaller and smaller. (Assume that the loop magically knows the best way to contract, and won't get snagged on jagged edges if it can possibly avoid them.) If the loop can always shrink all the way to a point, then the aquarium's interior is simply connected. If sometimes the loop gets caught — for example, around the central hole in the doughnut — then the object is not simply connected. For other uses, see Aquarium (disambiguation). ...

Notice that the definition only rules out "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 have no holes of any dimension, is called contractibility. In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i. ...

Formal definition and equivalent formulations

A topological space X is called simply connected if it is path-connected (which requires it to be a connected space) 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. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... Illustration of a unit circle. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, the open unit disk around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: The closed unit disk around P is the set of points whose distance from P is less than or equal...

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, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... 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 (elliptic) cylinder, the Möbius strip 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 (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... A torus. ... A right circular cylinder In mathematics, a cylinder is a quadric, i. ... 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 is one of the basic structures investigated in functional analysis. ... 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 a generalization of Euclidean space which is not restricted to finite dimensions. ... 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 n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... 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 "handles" of the surface. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... 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 &#8594; 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...

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, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...

• 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 line 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(u). 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 a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... 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. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... 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

• Deformation retract
• Unicoherent
• n-connected