Following this question Are there compact manifolds without boundary?
I'm asking if there is any example of a manifold with boundary that is not compact, Is the interval $X=[0,1)$ a counterexample ? I mean $X$ is a $1-$dimensional manifold with boundary $\partial X=\{0\}$ but $X$ is not compact because $X$ is not closed in $\mathbb R$. Is this correct and are there any other examples ?