Past exam of the options II course of the University of Oxford / 2010 / E / E3 / a

A subset $A$ is connected if its subspace topology admits no separation into two disjoint nonempty relatively open subsets. Equivalently, every continuous map $A\to \{0,1\}$, with the target discrete, is constant: a nonconstant map gives the separation by its two fibers, and a separation defines such a map.

The connected subsets of the usual real line are the empty set and all intervals, including singletons, bounded intervals of any endpoint type, rays, and the whole line. Equivalently, a nonempty subset is connected exactly when it contains every point between any two of its points.

Use the characterization by continuous maps to the discrete two-point space. Let $A={\bigcup }_i{A}_i$ and let $f:A\to \{0,1\}$ be continuous. Its restriction to each connected $A_i$ is constant. Since $A_i\cap A_j\ne \varnothing$, those two constants agree for every pair $i,j$. Hence $f$ is constant on their union. Thus the union is connected. If the indexing set is empty, the union is empty and is connected directly.