Editing Fundamental group (section)

===Homotopy of loops===

Given a topological space ''X'', a ''[[Loop (topology)|loop]] based at <math>x_0</math>'' is defined to be a [[continuous function (topology)|continuous function]] (also known as a continuous map)
:<math>\gamma \colon [0, 1] \to X</math>
such that the starting point <math>\gamma(0)</math> and the end point <math>\gamma(1)</math> are both equal to <math>x_0</math>.

[[File:Homotopy_of_pointed_circle_maps.png|Homotopy of loops|thumb]]
A ''[[homotopy]]'' is a continuous interpolation between two loops. More precisely, a homotopy between two loops <math>\gamma, \gamma' \colon [0, 1] \to X</math> (based at the same point <math>x_0</math>) is a continuous map
:<math>h \colon [0, 1] \times [0, 1] \to X,</math>
such that
* <math>h(0, t) = x_0</math> for all <math>t \in [0, 1],</math> that is, the starting point of the homotopy is <math>x_0</math> for all ''t'' (which is often thought of as a time parameter). 
* <math>h(1, t) = x_0</math> for all <math>t \in [0, 1],</math> that is, similarly the end point stays at <math>x_0</math> for all ''t''.
* <math>h(r, 0) = \gamma(r),\, h(r, 1) = \gamma'(r)</math> for all <math>r \in [0, 1]</math>.

If such a homotopy ''h'' exists, <math>\gamma</math> and <math>\gamma'</math> are said to be ''homotopic''. The relation "<math>\gamma</math> is homotopic to <math>\gamma'</math>" is an [[equivalence relation]] so that the set of equivalence classes can be considered:
:<math>\pi_1(X, x_0) := \{ \text{all loops }\gamma \text{ based at }x_0 \} / \text{homotopy}</math>.
This set (with the group structure described below) is called the ''fundamental group'' of the topological space ''X'' at the base point <math>x_0</math>. The purpose of considering the equivalence classes of loops [[up to]] homotopy, as opposed to the set of all loops (the so-called [[loop space]] of ''X'') is that the latter, while being useful for various purposes, is a rather big and unwieldy object. By contrast the above [[quotient set|quotient]] is, in many cases, more  manageable and computable.