Editing Fundamental group (section)

=== The 2-sphere ===
[[File:P1S2all.jpg|upright=1.8|thumb|A loop on a [[2-sphere]] (the surface of a ball) being contracted to a point]]
A path-connected space whose fundamental group is trivial is called [[simply connected space|simply connected]]. 
For example, the [[2-sphere]] <math>S^2 = \left\{(x, y, z) \in \R^3 \mid x^2 + y^2 + z^2 = 1\right\}</math> depicted on the right, and also all the [[n-sphere|higher-dimensional spheres]], are simply-connected. The figure illustrates a homotopy contracting one particular loop to the constant loop. This idea can be adapted to all loops <math>\gamma</math> such that there is a point <math>(x, y, z) \in S^2</math> that is {{em|not}} in the image of <math>\gamma.</math> However, since there are loops such that <math>\gamma([0, 1]) = S^2</math> (constructed from the [[Peano curve]], for example), a complete [[mathematical proof|proof]] requires more careful analysis with tools from algebraic topology, such as the [[Seifert–van Kampen theorem]] or the [[cellular approximation theorem]].