Definition:Path (Topology)

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Let $I \subset \R$ be the closed real interval $\closedint a b$.

A path in $T$ is a continuous mapping $\gamma: I \to S$.

The mapping $\gamma$ can be referred as:
 * a path (in $T$) joining $\map \gamma a$ and $\map \gamma b$

or:
 * a path (in $T$) from $\map \gamma a$ to $\map \gamma b$.

It is common to refer to a point $z = \map \gamma t$ as a point on the path $\gamma$, even though $z$ is in fact on the image of $\gamma$.

Also see

 * Definition:Loop (Topology)
 * Definition:Contour