A <metric space> is <connected> when it has no <separation of a topological space>: it cannot be written as the union of two disjoint nonempty <open subsets>. Equivalently, the only <subsets> that are both <open> and <closed> are the <empty set> and the whole space.
$$
\boxed{X\text{ connected}\iff\text{no two disjoint nonempty open sets cover }X.}
$$
The <open sets> here are those of the <metric topology>.

Solution

Past exam of the metric spaces and complex analysis course of the University of Oxford / 2015 / 2 / a / i / Solution

 Solution (source code)