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The '''fundamental group''' is a basic construction of [[algebraic topology]]. For a given surface, such as a torus or even the (humorously named) dunce cap, the fundamental group is the set of loops in the space, but with two loops regarded as equivalent if they may be continuously deformed to one another, a relation known as [[homotopy]].

The '''fundamental group''' is a basic construction of [[algebraic topology]]. For a given surface, such as a torus or even the (humorously named) dunce cap, the fundamental group is the set of loops in the space that have the following characteristic:  each loop can be continuously deformed to one another, a relation known as [[homotopy]].

To a [[topological space]] <math>X</math>, the fundamental group construction associates an algebraic object which describes the set of "holes" in the space.   

To a [[topological space]] <math>X</math>, the fundamental group construction associates an algebraic object which describes the set of "holes" in the space.