Is a straight line closed in $\mathbb{R}^2$ with no interior points? Is a straight line closed in $\mathbb{R}^2$ with no interior points? 
It seems to me that all the points on a straight line are limit points. None of them can form a neighborhood of $ r>0 $ that is completely in the straight line. So a straight line is closed with no interior points in $\mathbb{R}^2$. Is that true? 

Yes, you are right.  Consider the line $L=\{(x,0):x\in\Bbb R\}$, i.e. the $x$-axis.  If let $P=(x,0)$, then the ball $B(P,\epsilon)$ for any $\epsilon>0$ will contain points with $y>0$ and $y<0$.  So, no $\epsilon$ ball is contained in the line.
