Editing Fundamental group (section)

===Gluing topological spaces===
Generalizing the statement above, for a family of path connected spaces <math>X_i,</math> the fundamental group <math display="inline">\pi_1 \left(\bigvee_{i \in I} X_i\right)</math> is the [[free product]] of the fundamental groups of the <math>X_i.</math><ref>{{harvtxt|May|1999|loc=Ch. 2, §8, Proposition}}</ref> This fact is a special case of the [[Seifert–van Kampen theorem]], which allows to compute, more generally, fundamental groups of spaces that are glued together from other spaces. For example, the 2-sphere <math>S^2</math> can be obtained by gluing two copies of slightly overlapping half-spheres along a [[neighborhood (mathematics)|neighborhood]] of the [[equator]]. In this case the theorem yields <math>\pi_1(S^2)</math> is trivial, since the two half-spheres are contractible and therefore have trivial fundamental group. The fundamental groups of surfaces, as mentioned above, can also be computed using this theorem.

In the parlance of category theory, the theorem can be concisely stated by saying that the fundamental group functor takes [[Pushout (category theory)|pushouts]] (in the category of topological spaces) along inclusions to pushouts (in the category of groups).<ref>{{harvtxt|May|1999|loc=Ch. 2, §7}}</ref>