Very basic question about submanifolds. I'm beginning to study differential geometry and I'm a litle confused about the concept of submanifold of a differentiable manifolds.
Can someone provide me an example of how to show that a non-open subset of a manifold is a submanifold and other showing it isn't?

Here are some very simple examples.
Let $M = \mathbb R^2$. Let $N$ be a straight line in $M$. Then $N$ is not open in $M$ and a submanifold of $M$. For example, you can choose global coordinates in $M$ such that $N$ is the first coordinate axis.
Let $P \subseteq M$ be the union of two straight lines which meet in one point. Then $P$ is not open in $M$, and $P$ isn't a submanifold of $M$. In fact, $P$ isn't even an manifold, because the meeting point of the two lines doesn't have a one-dimensional local trivialization.
