Suppose that $A$ is a connected subset of a space $X$ and that $A\subseteq B \subseteq \bar A$. Prove $B$ is connected. I think I can prove the closure of $A$, that is $\bar A$, is connected, as there are many other threads on this site.  I am then just not sure how to make the jump to show formally that B is connected.

If $B$ is not connected then non-empty sets $U,V$ exist with $U\cap\overline{V}=\varnothing=\overline{U}\cap V$ and $B=U\cup V$. 
Then the sets $U\cap A$ and $V\cap A$ are disjoint, open in $A$ and are covering $A$. 
So one of these sets must be empty since $A$ is connected. 
If $U\cap A=\varnothing$ then $A\subseteq V$ and consequently $U\subseteq B\subseteq \overline{A}\subseteq\overline{V}$. 
Then $\varnothing\neq U\subseteq\overline{V}$ contradicting that $U\cap\overline{V}=\varnothing$. 
Likewise $V\cap A=\varnothing$ leads to a contradiction.
