Path (topology)

A curve in a topological space $X$ is a continuous function $f\; :\; J\; \backslash to\; X$ from a non-empty and non-degenerate interval $J\; \backslash subseteq\; \backslash R.$

A {{em|path}} in $X$ is a curve $f\; :\; [a,\; b]\; \backslash to\; X$ whose domain $[a,\; b]$ is a compact non-degenerate interval (meaning are real numbers), where $f(a)$ is called the {{em|initial point}} of the path and $f(b)$ is called its {{em|terminal point}}.

A {{em|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 {{em|path}} is sometimes, especially in homotopy theory, defined to be a continuous function $f\; :\; [0,\; 1]\; \backslash to\; X$ from the closed unit interval $I\; :=\; [0,\; 1]$ into $X.$

{{anchor|Arc|C0 arc}}

An {{em|arc}} or {{mvar|C}}⁰{{em|-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 $x\; \backslash in\; X$ is a path from $x$ to $x.$ A loop may be equally well regarded as a map $f\; :\; [0,\; 1]\; \backslash to\; X$ with $f(0)\; =\; f(1)$ or as a continuous map from the unit circle $S^1$ to $X$

:$f\; :\; S^1\; \backslash to\; X.$

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 $X.$

{{Main|Homotopy}}

Image:Homotopy between two paths.svg

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 $f\_t\; :\; [0,\; 1]\; \backslash to\; X$ indexed by $I\; =\; [0,\; 1]$ such that

• $f\_t(0)\; =\; x\_0$ and $f\_t(1)\; =\; x\_1$ are fixed.
• the map $F\; :\; [0,\; 1]\; \backslash times\; [0,\; 1]\; \backslash to\; X$ 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 $g$:

:

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 {{em|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 $\backslash pi\_1\backslash left(X,\; x\_0\backslash right).$

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 $a\; \backslash geq\; 0.$ (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:

:

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 $\backslash left(|f|\; +\; |g|\; +\; |h|\backslash right)/2$ 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.

• {{Section link|Curve#Topology}}
• {{annotated link|Locally path-connected space}}
• Path space (disambiguation)
• {{annotated link|Path-connected space}}

• Ronald Brown, Topology and groupoids, Booksurge PLC, (2006).
• J. Peter May, A concise course in algebraic topology, University of Chicago Press, (1999).
• James Munkres, Topology 2ed, Prentice Hall, (2000).

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Category:Topology

Category:Homotopy theory