Connected subsets of $\mathbb{R}$

Interval connected. If $A$ is an interval and $A=U\cup V$ with $U,V$ disjoint nonempty open in $A$, take $u\in U$, $v\in V$, WLOG $u<v$. Since $A$ is an interval, $[u,v]\subseteq A$. The sets $U\cap[u,v]$ and $V\cap[u,v]$ give a separation of $[u,v]$, contradiction with connectedness of intervals.

Non-interval disconnected. If $a<b<c$ with $a,c\in A$, $b\notin A$, then $A\cap]-\infty,b[$ and $A\cap]b,+\infty[$ separate $A$.

Conclusion. Connected subsets of $\mathbb{R}$ = intervals.