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In mathematics, a path in a topological space ${\displaystyle X}$ is a continuous function from a closed interval into ${\displaystyle X.}$

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 ${\displaystyle X}$ is often denoted ${\displaystyle \pi _{0}(X).}$

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If ${\displaystyle X}$ is a topological space with basepoint ${\displaystyle x_{0},}$ then a path in ${\displaystyle X}$ is one whose initial point is ${\displaystyle x_{0}}$. Likewise, a loop in ${\displaystyle X}$ is one that is based at ${\displaystyle x_{0}}$.

Definition

A curve in a topological space ${\displaystyle X}$ is a continuous function ${\displaystyle f:J\to X}$ from a non-empty and non-degenerate interval ${\displaystyle J\subseteq \mathbb {R} .}$ A path in ${\displaystyle X}$ is a curve ${\displaystyle f:\to X}$ whose domain ${\displaystyle }$ is a compact non-degenerate interval (meaning ${\displaystyle a<b}$ are real numbers), where ${\displaystyle f(a)}$ is called the initial point of the path and ${\displaystyle f(b)}$ is called its terminal point. A path from ${\displaystyle x}$ to ${\displaystyle y}$ is a path whose initial point is ${\displaystyle x}$ and whose terminal point is ${\displaystyle y.}$ Every non-degenerate compact interval ${\displaystyle }$ is homeomorphic to ${\displaystyle ,}$ which is why a path is sometimes, especially in homotopy theory, defined to be a continuous function ${\displaystyle f:\to X}$ from the closed unit interval ${\displaystyle I:=}$ into ${\displaystyle X.}$ An arc or C⁰-arc in ${\displaystyle X}$ is a path in ${\displaystyle X}$ that is also a topological embedding.

Importantly, a path is not just a subset of ${\displaystyle X}$ that "looks like" a curve, it also includes a parameterization. For example, the maps ${\displaystyle f(x)=x}$ and ${\displaystyle g(x)=x^{2}}$ represent two different paths from 0 to 1 on the real line.

A loop in a space ${\displaystyle X}$ based at ${\displaystyle x\in X}$ is a path from ${\displaystyle x}$ to ${\displaystyle x.}$ A loop may be equally well regarded as a map ${\displaystyle f:0,1\to X}$ with ${\displaystyle f(0)=f(1)}$ or as a continuous map from the unit circle ${\displaystyle S^{1}}$ to ${\displaystyle X}$

${\displaystyle f:S^{1}\to X.}$

This is because ${\displaystyle S^{1}}$ is the quotient space of ${\displaystyle I=0,1}$ when ${\displaystyle 0}$ is identified with ${\displaystyle 1.}$ The set of all loops in ${\displaystyle X}$ forms a space called the loop space of ${\displaystyle X.}$

Homotopy of pathsedit

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 ${\displaystyle X}$ is a family of paths ${\displaystyle f_{t}:0,1\to X}$ indexed by ${\displaystyle I=0,1}$ such that

• ${\displaystyle f_{t}(0)=x_{0}}$ and ${\displaystyle f_{t}(1)=x_{1}}$ are fixed.
• the map ${\displaystyle F:0,1\times 0,1\to X}$ given by ${\displaystyle F(s,t)=f_{t}(s)}$ is continuous.

The paths ${\displaystyle f_{0}}$ and ${\displaystyle f_{1}}$ 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 ${\displaystyle f}$ under this relation is called the homotopy class of ${\displaystyle f,}$Zdroj:https://en.wikipedia.org?pojem=Path_(topology)
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