 Simply connected space

In topology, a topological space is called simply connected (or 1connected) if it is pathconnected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question (see below for an informal discussion).
If a space is not simply connected, it is convenient to measure the extent to which it fails to be simply connected; this is done by the fundamental group. Intuitively, the fundamental group measures how the holes behave on a space; if there are no holes, the fundamental group is trivial — equivalently, the space is simply connected.
Contents
Informal discussion
Informally, a thick 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 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 oldfashioned term, multiply connected.
To illustrate the notion of simple connectedness, suppose we are considering an object in three dimensions; for example, an object in the shape of a box, a doughnut, or a corkscrew. 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.
Notice that the definition only rules out "handleshaped" 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.
Formal definition and equivalent formulations
A topological space X is called simply connected if it is pathconnected and any continuous map f : S^{1} → X (where S^{1} denotes the unit circle in Euclidean 2space) can be contracted to a point in the following sense: there exists a continuous map F : D^{2} → X (where D^{2} denotes the unit disk in Euclidean 2space) such that F restricted to S^{1} is f.
An equivalent formulation is this: X is simply connected if and only if it is pathconnected, 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.
A third way to express the same: X is simply connected if and only if X is pathconnected and the fundamental group of X at each of its points is trivial, i.e. consists only of the identity element.
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.
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 X 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 connected extended complement exactly when each of its connected components are simply connected.
Examples
 The Euclidean plane R^{2} is simply connected, but R^{2} minus the origin (0,0) is not. If n > 2, then both R^{n} and R^{n} minus the origin are simply connected.
 Analogously: the ndimensional sphere S^{n} is simply connected if and only if n ≥ 2.
 Every convex subset of R^{n} is simply connected.
 A torus, the (elliptic) cylinder, the Möbius strip, the projective plane 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).
 Similarly, the onepoint compactification of R is not simply connected (even though R is simply connected).
Properties
A surface (twodimensional 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.
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.
If X and Y are homotopy equivalent and X is simply connected, then so is Y.
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 simple connectedness is important in complex analysis because of the following facts:
 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 nonempty open simply connected subset of C (except for C itself) is conformally equivalent to the unit disk.
The notion of simply connectedness is also a crucial condition in the Poincaré lemma.
See also
 Deformation retract
 nconnected
 Pathconnected
 Unicoherent
References
 Spanier, Edwin (December 1994). Algebraic Topology. Springer. ISBN 0387944265.
 Conway, John (1986). Functions of One Complex Variable I. Springer. ISBN 0387903283.
 Bourbaki, Nicolas (2005). Lie Groups and Lie Algebras. Springer. ISBN 3540434054.
 Gamelin, Theodore (January 2001). Complex Analysis. Springer. ISBN 0387950699.
 Joshi, Kapli (August 1983). Introduction to General Topology. New Age Publishers. ISBN 0852264445.
Categories: Algebraic topology
 Connection (mathematics)
 Properties of topological spaces
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