proof that a path connected space is connected

Let $X$ be a path connected topological space. Suppose that $X=A\cup B$, where $A$ and $B$ are non empty, disjoint, open sets. Let $a\in A$, $b\in B$, and let $\gamma :I\to X$ denote a path from $a$ to $b$.

We have $I={\gamma }^{-1}(A)\cup {\gamma }^{-1}(B)$, where ${\gamma }^{-1}(A),{\gamma }^{-1}(B)$ are non empty, open and disjoint. Since $I$ is connected, this is a contradiction, which concludes the proof.