Definition:Path (Topology)

Let $$T$$ be a topological space.

Let $$I \subset \R$$ be the closed real 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 $$\gamma$$.