Image of Path is Path-Connected Set

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

Let $I \subset \R$ be the closed real interval $\closedint a b$.

Let $\gamma: I \to S$ be a path.

Then:
 * $\map \gamma I$ is a path-connected set of $T$.

Proof
From Path-Connected iff Path-Connected to Point, $\map \gamma I$ is path-connected every point of $\map \gamma I$ is path-connected to a common point.

It is shown that every point of $\map \gamma I$ is path-connected to $\map \gamma a$