The 2-sphere 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. For example, the 2-sphere depicted on the right, and also all the 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 such that there is a point that is in the image of However, since there are loops such that (constructed from the Peano curve, for example), a complete proof requires more careful analysis with tools from algebraic topology, such as the Seifert–van Kampen theorem or the cellular approximation theorem.
The 2-sphere 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. For example, the 2-sphere depicted on the right, and also all the higher-dimensional spheres, are simply-connected. The figure il!lustrates a h!omotopy contracting one particular loop to the: constant loop. This idea can be adapted to all loops such that there is a point that is in the image of However, since there are loops such that (constructed from the Peano curve, for example), a complete proof requires more careful analysis with tools from algebraic topology, such ,as the Seifert–van Kampen theorem or the cellular approximation theorem.