PATH (TOPOLOGY)

In mathematics, a 'path' in a topological space ''X'' is a continuous map ''f'' from the unit interval ''I'' = [0,1] to ''X''
:''f'' : ''I'' → ''X''.
The ''initial point'' of the path is ''f''(0) and the ''terminal point'' is ''f''(1). One often speaks of a "path from ''x'' to ''y''" where ''x'' and ''y'' are the initial and terminal points of the path. Note that a path is not just a subset of ''X'' which "looks like" a curve, it also includes a parametrization. For example, the maps ''f''(''x'') = ''x'' and ''g''(''x'') = ''x''² represent two different paths from 0 to 1 on the real line.
A 'loop' in a space ''X'' based at ''x'' ∈ ''X'' is a path from ''x'' to ''x''. A loop may be equally well regarded as a map ''f'' : ''I'' → ''X'' with ''f''(0) = ''f''(1) or as a continuous map from the unit circle ''S''¹ to ''X''
:''f'' : ''S''¹ → ''X''.
This is because ''S''¹ may be regarded as a quotient of ''I'' under the identification 0 ∼ 1. The set of all loops in ''X'' forms a space called the loop space of ''X''.
A topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into a set of path-connected components. The set of path-connected components of a space ''X'' is often denoted π₀(''X'');.
One can also define paths and loops in pointed spaces, which are important in homotopy theory. If ''X'' is a topological space with basepoint ''x''₀, then a path in ''X'' is one whose initial point in ''x''₀. Likewise, a loop in ''X'' is one that is based at ''x''₀.

Contents

Homotopy of paths

Path composition

Fundamental groupoid

Homotopy of paths

Paths and loops are extremely important in 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.
Specifically, a homotopy of paths in ''X'' is a family of paths ''f''_''t'' : ''I'' → ''X'' such that

★ ''f''_''t''(0) = ''x''₀ and ''f''_''t''(1) = ''x''₁ are fixed.

★ the map ''F'' : ''I'' × ''I'' → ''X'' given by ''F''(''s'', ''t'') = ''f''_''t''(''s'') is continuous.
The paths ''f''₀ and ''f''₁ connected by a homotopy are said to 'homotopic'. One can likewise define a homotopy of loops keeping the base point fixed.
The property of being homotopic defines an equivalence relation on paths in a topological space. The equivalence class of a path ''f'' under this relation is called the 'homotopy class' of ''f'', often denoted [''f''].

Path composition

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'':
:
Clearly path composition is only defined when the terminal point of ''f'' coincides with the initial point of ''g''. If one considers all loops based at a point ''x''₀, then path composition is a binary operation.
Path composition, whenever defined, is not associative due to the difference in parametrization. It ''is'' associative at the level of homotopy however. That is, [(''fg'')''h''] = [''f''(''gh'')]. Path composition defines a group structure on the set of homotopy classes of loops based at a point ''x''₀ in ''X''. The resultant group is called the fundamental group of ''X'' based at ''x''₀, usually denoted π₁(''X'',''x''₀).

Fundamental groupoid

There is a categorical picture of paths which is sometimes useful. Any topological space ''X'' can be viewed as 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 ''x''₀ in ''X'' is just the fundamental group based at ''X''.