Definition:Path (Topology)

Definition
Let $T$ be a topological space.

Let $I \subset \R$ be the unit interval $\left[{0 \,.\,.\, 1}\right]$.

Let $a, b \in T$.

A path from $a$ to $b$ is a continuous mapping $f: I \to T$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.

The mapping $f$ can be described as a path (in $T\,$) joining $a$ and $b$.

Alternative Definition
This definition is the one usually given in the field of complex analysis, but is still completely relevant in the context of topology.

Let $T$ be a topological space.

Let $I \subset \R$ be the closed real interval $\left[{a \,.\,.\, b}\right]$.

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

Its initial point is $\gamma \left({a}\right)$ and its final point is $\gamma \left({b}\right)$.

The mapping $\gamma$ can be described as a path (in $T \ $) joining $\gamma \left({a}\right)$ and $\gamma \left({b}\right)$, or a path from $\gamma \left({a}\right)$ to $\gamma \left({b}\right)$.

It is common to refer to a point $z = \gamma \left({t}\right)$ as a point on the path $\gamma$, even though $z$ is in fact on the image of $\gamma$.