\begin{definition}[Definition:Path-Connected/Topology/Points]
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $a, b \in S$ be such that there exists a path from $a$ to $b$.
That is, there exists a continuous mapping $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.
Then $a$ and $b$ are '''path-connected in $T$'''.
\end{definition}