linguistic:paraphrasing
The 2-sphere

A path-connected space whose fundamental group is trivial is called simply connected. For example, the 2-sphere S 2 = { (x, y, z) ∈ R 3 ∣ x 2 + y 2 + z 2 = 1 } {\displaystyle S^{2}=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}} 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 γ {\displaystyle \gamma } such that there is a point (x, y, z) ∈ S 2 {\displaystyle (x,y,z)\in S^{2}} that is not in the image of γ. {\displaystyle \gamma.} However, since there are loops such that γ () = S 2 {\displaystyle \gamma ()=S^{2}} (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.