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? 

Let denote $L$ a line in $\mathbb R^2$. We have that $$\forall x\in L,\forall \varepsilon>0,\quad B_x(\varepsilon)\cap L\neq\emptyset\quad\text{and}\quad B_x(\varepsilon)\cap L^c\neq\emptyset$$
where $B_x(\varepsilon)=\{y\in\mathbb R^2\mid \|x-y\|<\varepsilon\}$. Therefore $$L=\partial L,$$ and thus $L$ is closed with no interior point.
In fact, you can prove that if $(V,\|\cdot \|)$ is a vector space such that $\dim V=n$, then all subspace $W$ such that $\dim W\leq n-1$ is closed with no interior point in $V$.
