Editing Simply connected space (section)

==Properties==
A surface (two-dimensional topological [[manifold]]) is simply connected if and only if it is connected and its [[Genus (mathematics)|genus]] (the number of {{em|handles}} of the surface) is 0.

A universal cover of any (suitable) space <math>X</math> is a simply connected space which maps to <math>X</math> via a [[covering map]].

If <math>X</math> and <math>Y</math> are [[homotopy equivalent]] and <math>X</math> is simply connected, then so is <math>Y.</math>

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 <math>\Complex \setminus \{ 0 \},</math> 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]] states that if <math>U</math> is a simply connected open subset of the [[Complex number|complex plane]] <math>\Complex,</math> and <math>f : U \to \Complex</math> is a [[holomorphic function]], then <math>f</math> has an [[Antiderivative (complex analysis)|antiderivative]] <math>F</math> on <math>U,</math> and the value of every [[line integral]] in <math>U</math> with integrand <math>f</math> depends only on the end points <math>u</math> and <math>v</math> of the path, and can be computed as <math>F(v) - F(u).</math> The integral thus does not depend on the particular path connecting <math>u</math> and <math>v,</math> 
* The [[Riemann mapping theorem]] states that any non-empty open simply connected subset of <math>\Complex</math> (except for <math>\Complex</math> itself) is [[Conformal map|conformally equivalent]] to the [[unit disk]].

The notion of simple connectedness is also a crucial condition in the [[Poincaré conjecture]].