Is it true that all $k$-submanifolds of a $m$-manifold are open subsets of some closed $k$-submanifold?

Question

Let $M$ be a $m$-dimensional (smooth) manifold. I know that $m$-submanifolds of $M$ are exactly the open subsets of $M$. Is it true that all $k$-submanifolds of $M$ are open subsets of some closed $k$-submanifold of $M$?

Answer 1

No. For example, in $\mathbb R^2$, the union of the $x$-axis and the $y$-axis with the origin removed is a $1$-submanifold. Any closed submanifold containing it would have to contain the origin. But there's no $1$-submanifold that contains the origin and both axes.