The '''fundamental group''' is a basic construction of [[algebraic topology]]. For a given surface, such as a torus or even (humorously) a dunce cap, its fundamental group is its set of path loops that has a useful characteristic 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. The fundamental group can be used is useful to distinguish between different topological spaces: for example, to prove rigorously that a surface with two holes cannot be smoothly deformed into a surface with three holes, one can check that these two spaces have different fundamental groups and so must be distinct. The fundamental group captures information about the set of holes in a space by looking at the set of loops drawn on the space.
== Simply Connected Spaces ==