Definition:Path (Topology)

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 $$b$$.