Union of connected space 
If $\{A_\alpha, \alpha \in I\}$ is a collection of conected spaces and $\cap A_\alpha \neq \emptyset$ then $\cup A_\alpha$ is connected.

My proof: If $\cup A_\alpha$ is not connected then we can find $V,W\neq\emptyset$ such that $V\cup W = \cup A_\alpha$ and $V\cap W = \emptyset$. Suppose $\cup A_\alpha$ is not connected. Since $\cap A_\alpha \neq \emptyset$ then $\exists x\in \cap A_\alpha$. Because $V\cup W = \cup A_\alpha$ and $V\cap W = \emptyset$ $x\in V$ or $x\in W$, but if $x\in V$ then $x\notin W$ therefore $\exists \beta \in I$ such that $x\notin A_\beta$, likewise if $x\in W$. We reach a contradiction, thus $\cup A_\alpha$ is connected

Put $A = \cup_aA_a$ and assume $V$ and $W$ are two nonempty disjoint sets for which $V\cup W = A$. Take $x \in \cap_aA_a$. Surely $x \in A$, so either $x \in V$ or $x \in W$. Furthermore $x \in A_a$ for all $a$, so either $A_a \subset V$ or $A_a \subset W$, since otherwise we could write $A_a$ as a disjoint union of $V$ and $W$. In fact if $A_a$ is a subset of either $V$ or $W$ for one $a$ then every $A_a$ in the collection is a subset of the same set, since $ \cap_aA_a \neq 0$ and they are all connected. Thus $A$ is either entirely contained in $V$ or $W$. Since we have assume $A = V \cup W$ this implies one of the two must be the empty set.
