Path (topology)

In mathematics, a path in a topological space \(X\) is a continuous function from a closed interval into \(X.\)

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 \(X\) is often denoted \(\pi_0(X).\)

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 is \(x_0\). Likewise, a loop in \(X\) is one that is based at \(x_0\).

A curve in a topological space \(X\) is a continuous function \(f : J \to X\) from a non-empty and non-degenerate interval \(J \subseteq \R.\) A path in \(X\) is a curve \(f : [a, b] \to X\) whose domain \([a, b]\) is a compact non-degenerate interval (meaning a < b\) are real numbers), where \(f(a)\) is called the initial point of the path and \(f(b)\) is called its terminal point. A path from \(x\) to \(y\) is a path whose initial point is \(x\) and whose terminal point is \(y.\) Every non-degenerate compact interval \([a, b]\) is homeomorphic to \([0, 1],\) which is why a path is sometimes, especially in homotopy theory, defined to be a continuous function \(f : [0, 1] \to X\) from the closed unit interval \(I := [0, 1]\) into \(X.\)

An arc or C-arc in \(X\) is a path in \(X\) that is also a topological embedding.

Importantly, a path is not just a subset of \(X\) that "looks like" a curve, it also includes a parameterization. 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 \(x \in X\) is a path from \(x\) to \(x.\) A loop may be equally well regarded as a map \(f : [0, 1] \to X\) with \(f(0) = f(1)\) or as a continuous map from the unit circle \(S^1\) to \(X\)

\(f : S^1 \to X.\)

This is because \(S^1\) is the quotient space of \(I = [0, 1]\) when \(0\) is identified with \(1.\) The set of all loops in \(X\) forms a space called the loop space of \(X.\)

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 \(X\) is a family of paths \(f_t : [0, 1] \to X\) indexed by \(I = [0, 1]\) such that

• \(f_t(0) = x_0\) and \(f_t(1) = x_1\) are fixed.
• the map \(F : [0, 1] \times [0, 1] \to X\) given by \(F(s, t) = f_t(s)\) is continuous.

The paths \(f_0\) and \(f_1\) connected by a homotopy are said to 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 \(f\) under this relation is called the homotopy class of \(f,\) often denoted \([f].\)

One can compose paths in a topological space in the following 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 \(g\):

\(fg(s) = \begin{cases}f(2s) & 0 \leq s \leq \frac{1}{2} \\ g(2s-1) & \frac{1}{2} \leq s \leq 1.\end{cases}\)

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. However it is associative up to path-homotopy. 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 \(\pi_1\left(X, x_0\right).\)

In situations calling for associativity of path composition "on the nose," a path in \(X\) may instead be defined as a continuous map from an interval \([0, a]\) to \(X\) for any real \(a \geq 0.\) (Such a path is called a Moore path.) A path \(f\) of this kind has a length \(|f|\) defined as \(a.\) Path composition is then defined as before with the following modification:

\(fg(s) = \begin{cases}f(s) & 0 \leq s \leq |f| \\ g(s-|f|) & |f| \leq s \leq |f| + |g|\end{cases}\)

Whereas with the previous definition, \(f,\) \(g\), and \(fg\) all have length \(1\) (the length of the domain of the map), this definition makes \(|fg| = |f| + |g|.\) What made associativity fail for the previous definition is that although \((fg)h\) and \(f(gh)\)have the same length, namely \(1,\) the midpoint of \((fg)h\) occurred between \(g\) and \(h,\) whereas the midpoint of \(f(gh)\) occurred between \(f\) and \(g\). With this modified definition \((fg)h\) and \(f(gh)\) have the same length, namely \(|f| + |g| + |h|,\) and the same midpoint, found at \(\left(|f| + |g| + |h|\right)/2\) in both \((fg)h\) and \(f(gh)\); more generally they have the same parametrization throughout.

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 \(x_0\) in \(X\) is just the fundamental group based at \(x_0\). 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 Van Kampen's Theorem.