Arc-Connected Space is Path-Connected

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space which is arc-connected.

Then $T$ is path-connected.

Proof
Let $T = \left({S, \tau}\right)$ be arc-connected.

Then $\forall x, y \in S$, there exists a continuous injection $f: \left[{0 \,.\,.\, 1}\right] \to S$, such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.

As $f$ is a continuous injection, it is also simply a continuous mapping.

The result follows from the definition of path-connectedness.