Path (topology)

In mathematics, a path in a topological space X is a continuous function f from the unit interval I = to XPaths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.One can compose paths in a topological space in an obvious manner. Suppose f is a path from x to y and g is a path from y to z. The path fg is defined as the path obtained by first traversing f and then traversing g:There is a categorical picture of paths which is sometimes useful. Any topological space X gives rise to a category where the objects are the points of X and the morphisms are the homotopy classes of paths. Since any morphism in this category is an isomorphism this category is a groupoid, called the fundamental groupoid of X. Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point x0 in X is just the fundamental group based at x0. More generally, one can define the fundamental groupoid on any subset A of X, using homotopy classes of paths joining points of A. This is convenient for the Van Kampen's Theorem.