Let \( X \) be a topological space. A [...] of \( X \) is a pair \( U, V \) of disjoint nonempty open subsets of \( X \) whose union is \( X \). The space \( X \) is said to be connected iff [...].
This definition is from Munkres. Leinster's definition feels a little less intuitively concrete.
Connectedness, formulation 2 (or now a lemma)
A space \( X \) is connected iff the only subsets of \( X \) that [...] are [...].
Again, from Munkres.