Spivak's definition of a $k$-dimensional mainfold

Question

In Calculus on Manifolds Spivak gives the following definition of a manifold:

But $\Bbb{R}^k \times \{0\}$ is a set of $(k+1)$-tuples and $V$ is a set of $n$-tuples, so $$V \cap (\Bbb{R}^k \times \{0\}) = \emptyset $$ if $k+1 \neq n$. This doesn't add up with what Spivak states it's equal to.

Shouldn't it be $\Bbb{R}^k \times \{0\}^{n - k}$?

Answer 1

Here in the expression $\mathbb R^k \times \{0\}$ the symbol $0$ denotes the zero vector in $\mathbb R^{n-k}$. Often the same symbol $0$ is used to mean different things. The description $\{ y \in V \mid y^{k+1} = \cdots = y^n = 0 \}$ might be more clear.