= Solution
Suppose a <path-connected space> had a <separation> $X=U\cup V$. Choose $u\in U$, $v\in V$, and a <path> $\gamma$ from $u$ to $v$. The <inverse images> $\gamma^{-1}(U)$ and $\gamma^{-1}(V)$ are disjoint <open sets> in the <subspace topology> of $[0,1]$, cover it, and are nonempty because they contain its opposite endpoints. This contradicts the <connectedness> of an <interval>.
Therefore \b[every <path-connected space> is <connected>]. Notice that it is the <subspace topology> on $[0,1]$ that is used in the inverse-image argument.
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