Arc-Connected Space is Path-Connected

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

Then $T$ is path-connected.

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

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

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

The result follows from definition of path-connected.