= Solution
A subset is <connected> when it has no separation into two disjoint nonempty relatively open sets. It is <path-connected> when every pair of its points can be joined by a <continuous> path from $[0,1]$ lying in the subset.
Path-<connectedness> implies <connectedness>; the reverse fails in general, for example for the <topologist's sine curve>. They are equivalent for open subsets of every <Euclidean space> $\mathbb R^n$. More generally this holds for open subsets of <locally path-connected spaces>.
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