Simply connected space Totally Explained

Everything about Simply Connected totally explained

In topology, a geometrical object or space is called simply connected (or 1-connected) if it's 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 isn't 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. To illustrate the notion of simple connectedness, suppose we're 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 "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.

Formal definition and equivalent formulations

A topological space X is called simply connected if it's 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.
An equivalent formulation is this: X is simply connected if and only if it's path-connected, and whenever p : [0,1] → X and q : [0,1] → X are two paths (for example: continuous maps) with the same start and endpoint (p(0) = q(0) and p(1) = q(1)), then p and q are homotopic relative, which clearly isn't simply connected.
The notion of simply 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 doesn't 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.Further Information

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