= Solution
A <topological space> is <connected> if it has no <separation>: two disjoint nonempty <open sets> whose union is the whole space. Equivalently, its only <subsets> that are both <open> and <closed> are the <empty set> and the whole space.
A <subset> is <connected> when its <subspace topology> has this property. Thus the sets in a proposed <separation> of a subset need only be relatively <open>, rather than <open> in the ambient space.
$$
\boxed{A\text{ connected}\iff A\text{ has no separation in its subspace topology}.}
$$
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