Path (topology)

In mathematics, a path in a topological space formula_1 is a continuous function from the closed unit interval formula_2 into formula_3 
Paths play an important role in the fields of Topology and Mathematical analysis. 
For example, 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 path-connected components. The set of path-connected components of a space formula_1 is often denoted formula_5 
One can also define paths and loops in pointed spaces, which are important in homotopy theory. If formula_1 is a topological space with basepoint formula_7 then a path in formula_1 is one whose initial point is formula_9. Likewise, a loop in formula_1 is one that is based at formula_9.
Definition.
A in a topological space formula_1 is a continuous function formula_13 from a non-empty and non-degenerate interval formula_14 
A ' in a formula_1 is a curve formula_16 whose domain formula_17 is a compact non-degenerate interval (meaning formula_18 are real numbers), where formula_19 is called the ' of the path and formula_20 is called its . 
A is a path whose initial point is formula_21 and whose terminal point is formula_22 
Every non-degenerate compact interval formula_17 is homeomorphic to formula_24 which is why a is sometimes, especially in homotopy theory, defined to be a continuous function formula_25 from the closed unit interval formula_26 into formula_3 
An ' or ' in formula_1 is a path in formula_1 that is also a topological embedding. 
Importantly, a path is not just a subset of formula_1 that "looks like" a curve, it also includes a parameterization. For example, the maps formula_31 and formula_32 represent two different paths from 0 to 1 on the real line. 
A loop in a space formula_1 based at formula_34 is a path from formula_21 to formula_36 A loop may be equally well regarded as a map formula_25 with formula_38 or as a continuous map from the unit circle formula_39 to formula_1
This is because formula_39 is the quotient space of formula_43 when formula_44 is identified with formula_45 The set of all loops in formula_1 forms a space called the loop space of formula_3
Homotopy of paths.
Paths 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.
Specifically, a homotopy of paths, or path-homotopy, in formula_1 is a family of paths formula_49 indexed by formula_43 such that
The paths formula_55 and formula_56 connected by a homotopy are said to be homotopic (or more precisely path-homotopic, to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.
The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path formula_57 under this relation is called the homotopy class of formula_58 often denoted formula_59 
Path composition.
One can compose paths in a topological space in the following manner. Suppose formula_57 is a path from formula_21 to formula_62 and formula_63 is a path from formula_62 to formula_65. The path formula_66 is defined as the path obtained by first traversing formula_57 and then traversing formula_63:
Clearly path composition is only defined when the terminal point of formula_57 coincides with the initial point of formula_71 If one considers all loops based at a point formula_7 then path composition is a binary operation. 
Path composition, whenever defined, is not associative due to the difference in parametrization. However it associative up to path-homotopy. That is, formula_73 Path composition defines a group structure on the set of homotopy classes of loops based at a point formula_9 in formula_3 The resultant group is called the fundamental group of formula_1 based at formula_7 usually denoted formula_78 
In situations calling for associativity of path composition "on the nose," a path in formula_1 may instead be defined as a continuous map from an interval formula_80 to formula_1 for any real formula_82 A path formula_57 of this kind has a length formula_84 defined as formula_85 Path composition is then defined as before with the following modification:
Whereas with the previous definition, formula_58 formula_63, and formula_66 all have length formula_90 (the length of the domain of the map), this definition makes formula_91 What made associativity fail for the previous definition is that although formula_92 and formula_93have the same length, namely formula_94 the midpoint of formula_92 occurred between formula_63 and formula_97 whereas the midpoint of formula_93 occurred between formula_57 and formula_63. With this modified definition formula_92 and formula_93 have the same length, namely formula_103 and the same midpoint, found at formula_104 in both formula_92 and formula_93; more generally they have the same parametrization throughout.
Fundamental groupoid.
There is a categorical picture of paths which is sometimes useful. Any topological space formula_1 gives rise to a category where the objects are the points of formula_1 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 formula_3 Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point formula_9 in formula_1 is just the fundamental group based at formula_9. More generally, one can define the fundamental groupoid on any subset formula_113 of formula_114 using homotopy classes of paths joining points of formula_115 This is convenient for the Van Kampen's Theorem.