TITLE: Is a straight line closed in $\mathbb{R}^2$ with no interior points?
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: Another way to see that it is closed is to check that the complement is open: Around any point not on the line, you can find an open ball not intersecting the line (e.g. the ball with radius half the distance of the point from the line).
And of course, an open ball around a point of the line will always contain points both from the line and from its complement, therefore every point of the line is a boundary point and its interior is indeed empty.