Can a single point in a manifold be seen as a sub manifold?

Question

In Pollack's differential topology, in Transversality, p.28, it reduced the study of the submanifold $Z$ to the simpler case, where $Z$ is a single point. But by the definition of manifold, it seems that a single point is not a submanifold.

Answer 1

As a commenters said, a single point is a $0$-dimensional manifold. This is consistent with the fact that $\mathbb{R}^0=\{0\}$, and also with the fact that the boundary of a $d$-dimensional manifold-with-boundary is a $(d-1)$-dimensional manifold. (The boundary of $[0,1]$ is $\{0,1\}$.)

Since there is just one topology on a single-point space, it is always a submanifold whenever it's contained in another manifold: the restriction topology coming from the ambient space agrees with the intrinsic topology of the space.