Path-connected spaces are connected

Statement

Let $X$ be a path-connected space. Then $X$ is connected.

Remarks

Proof: suppose open sets $U$ and $V$ split $X$, with $x \in U$ and $y \in V$. Take a path $\gamma$ from $x$ to $y$. Then $\gamma([0, 1])$ is connected, by continuous image of a connected set is connected and connectedness of intervals. But $U$ and $V$ split it, a contradiction. The converse fails, as the topologist's sine curve shows. It holds for open subsets of R^n.

Explanations

L1 human

If you can walk between any two points, the space cannot be split into two separate open parts.