Path in a Metric Space¶

Definition¶

In a metric space $(X, d_{X})$ , a path is a continuous function $p\colon [0,1] \to X$ $X$ . If for two points $a, b \in X$ $X$ , the path $p$ satisfies $p(0) = a$ and $p(1) = b$ $b$ , it is termed as a path from $a$ to $b$ .

Interpretation¶

A path is a continuous journey from one point to another within the space: the parameter $t$ runs through $[0,1]$ like time, and $p(t)$ is the position at time $t$ . It is like drawing a curve without lifting your pencil, with the curve entirely contained in the space.

The interval from 0 to 1 with points 0, t and 1, mapped by p onto a curve inside a space X that runs from a = p(0) through p(t) to b = p(1) without jumps.

The interval $[0,1]$ on the left is the domain of $p$ , and the red curve in $X$ is its image. The start $p(0)=a$ and the end $p(1)=b$ are the endpoints of the curve, and an intermediate time $t$ is sent to the point $p(t)$ on the way. Continuity of $p$ means that close times give close positions, so the curve has no jumps.

A path is the function $p$ , not only the curve it traces. Different paths can trace the same curve: $t\mapsto p(t^2)$ runs through the same points as $p$ , at a different speed. A path may also stop, run back over itself or be constant, and it may have corners, since only continuity is required, not differentiability.

Paths define path-connectedness, and they reduce questions about a set to questions on $[0,1]$ : for a continuous real-valued $f$ , the composition $f\circ p$ is a continuous function of one variable. This is how the intermediate value theorem on connected sets can be proved for path-connected sets. Math IV studies differentiable curves through their velocity vectors and the chain rule on a curve.

Key Results¶

If $p$ is a path from $a$ to $b$ and $q$ is a path from $b$ to $c$ , then it is possible to construct a path from $a$ to $c$ using $p$ and $q$ . This is called the concatenation.
For any path $p$ from $a$ to $b$ , the reverse path $p(1-t)$ is a path from $b$ to $a$ .

Examples¶

In Euclidean space, the function $\boldsymbol{p}(t)=(1-t)\boldsymbol{a}+t\boldsymbol{b}$ is a path from $\boldsymbol{a}$ to $\boldsymbol{b}$ .
The function $\boldsymbol{p}(t)=(\cos(2\pi t),\sin(2\pi t))$ is a path from $(1,0)$ back to itself (a loop).

Prerequisites¶

Continuity between Metric Spaces

Sources¶

Math III: Analysis · Lecture 10

Symbols: element of, pi

Last updated: 20 October 2026 03:02