= Solution
{c}
A <connected set> is one whose relative topology admits no decomposition into two disjoint nonempty open sets. A <path-connected> set permits, between any two points, a continuous map $\gamma:[0,1]\to A$ with the prescribed endpoints.
Path connectedness implies connectedness, because a continuous image of an interval is connected. The converse can fail, for example for the <topologist's sine curve>. On open subsets of Euclidean spaces $\mathbb R^n$, the two properties are equivalent; more generally this holds in locally path-connected spaces.
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