Why is this space locally connected? 
This is from an old paper of FB Jones. 
Unless I am mistaken, the space is a countable union of closed nowhere dense sets (it is dense in $\mathbb R ^3$ and is a countable union of arcs). Is this correct?
My main question is, why is the example locally connected?  I'm having a hard time picturing a connected open neighborhood.
 A: 
Unless I am mistaken, the space is a countable union of closed nowhere dense sets (it is dense in $\mathbb R ^3$ and is a countable union of arcs). Is this correct?

Yes, that is correct.

My main question is, why is the example locally connected?  I'm having a hard time picturing a connected open neighborhood.

Let the space be called $X$.  I claim that if $U\subset\mathbb{R}^3$ is open and convex, then $U\cap X$ is connected.  Indeed, let $x,y\in U\cap X$; I will show that any clopen subset $V$ of $U\cap X$ which contains $x$ must also contain $y$.  Since $V$ is closed, it suffices to show $V$ intersects any $U_n\subseteq U$ which contains $y$.  So suppose $y\in U_n\subseteq U$.  Since $V$ is open in $X$, there is some $m$ such that $V$ contains $U_m\cap X$.  By construction, $X$ then contains the line segment from some point of $U_m$ to some point of $U_n$.  Since $U$ is convex, that entire line segment is also contained in $U$.  Since $V$ is clopen in $U\cap X$ and the line segment is connected, $V$ must also contain the entire line segment.  Thus $V$ contains a point of $U_n$, as desired.
