Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / E / E3 / a

A subset $A$ is connected if, with its subspace topology, it cannot be written as $A=U\cup V$ with $U,V$ disjoint nonempty relatively open sets. Equivalently its only subsets both open and closed relative to $A$ are $\varnothing ,A$. This definition includes the empty set and singletons as connected sets.

The connected subsets of the real line are exactly the intervals, including singletons, unbounded intervals and the empty set. Equivalently, whenever $x<y$ belong to the set, every $z$ with $x<z<y$ also belongs to it. Endpoints may or may not be included.

Use the separation definition in (i). If $A={\bigcup }_i{A}_i$ had a separation $A=U\cup V$, the intersections with each connected $A_i$ show that $A_i$ lies wholly in one side. Choose one nonempty $A_{i_0}$; every other $A_i$ intersects it and must lie on that same side. Thus the other side is empty, a contradiction. If the union is empty it is connected by definition. Therefore $\boxed{\bigcup _i{A}_i\text{ is connected}}$.