I have defined a submanifold as:
$M \subset \mathbb{R^n}$ is called a $k$-dim submanifold of $\mathbb{R^n}$ if $\forall x \in M$ the following condition holds:
$\exists U \subset \mathbb{R^n}$ open, $x \in U$, $V \subset \mathbb{R^n}$ open with $h: U \to V$ a diffeomorphism such that:
$h(U \cap M) = V \cap (\mathbb{R^k} \times \{0 \}^{n-k}) = \{ y \in V : y^{k+1} = \dotsb = y^n = 0 \}$
Now I'm not really sure I understand this definition, so I tried to look at some examples to help myself understand it.
I read that a point in $\mathbb{R^n}$ is a 0-dimensional submanifold, and any open set is a n-dimensional submanifold.
I'm trying to use the definition of a submanifold to show that any point is a 0-dimensional submanifold:
Let $M = \{ a \}$ this is called a 0-dimensional submanifold if $ \forall x \in M$ satisfies $h(U \cap M ) = V \cap (\mathbb{R^0} \times \{0\}^n)$ I'm having troubles understanding "$ (\mathbb{R^0} \times \{0\}^n)$ ". Moreover, if we had a random set $M$ how can we find out if it is a $k$ submanifold? How would we find the dimension (with the point I was told it is a 0 dimensional manifold, but I don't know if I could have guessed it).
Thanks.