I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions.
He defines a manifold as some space $M$ that satisfies:
$(M)$: $\forall x \in M, \exists U,V \subset \mathbb{R}^n$, (both open), with $x\in U$ and a diffeomorphism $h: U\rightarrow V$ such that $h(U \cap V)=V\cap (\mathbb{R}^k \times \{0\})=\{y\in V | y^{k+1}=\dots=y^n=0 \}$
He then defines a manifold with boundary with the similar definition:
Let $\mathbb{H}^k=\{x\in \mathbb{R}^k|x^k\geq 0\}$
$(M')$: $\forall x \in M, \exists U,V \subset \mathbb{R}^n$, (both open), with $x\in U$ and a diffeomorphism $h: U\rightarrow V$ such that $h(U\cap M)=V\cap (\mathbb{H}^k \times \{0\})=\{y\in V | y^k \geq 0, y^{k+1}=\dots=y^n=0\}$
He then goes on to say that "It is important to note that conditions $(M)$ and $(M')$ cannot both hold for the same $x$"...However, isn't $\mathbb{H}^k\subset\mathbb{R}^k$, so of course if $(M')$ holds, wouldn't $(M)$ hold as well? Also why does this boundary have to have a positive $k^{th}$ component?
Is this $\partial M$ the intersection between $M$ and $\{y\in \mathbb{H}^k | y^k=0\}\subset \mathbb{H}^k$? If we consider a finite line in $\mathbb{R}^2$ as a one dimensional manifold, then the ends would surely be its boundary...however I see no reason why $x_2$ (aka $y$) has to have zero values.