Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2011 / 4 / ii / Solution

A connected space $S$ cannot be expressed as $S=A\cup B$ with $A,B$ disjoint nonempty relatively open sets. Equivalently, the only subsets simultaneously relatively open and closed are $\varnothing$ and $S$. A decomposition into such $A,B$ is a separation of the topological space.