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 $B\subset U\cup V$ where $U$ and $V$ are disjoint, open sets, then the same thing can be said for $A$. But since $A$ is connected, then $U$ or $V$ is disjoint with $A$. Let's say WLOG that $A\cap U=\emptyset$.
Therefore, $A\subset \complement U$ and (since $\complement U$ is closed) $\bar A\subset \complement U$. This implies that $B\subset\complement U$, or equivalently, $B\cap U=\emptyset$.
