Why did Spivak choose this definitions? ("Calculus on Manifolds", the definition of the norm, the definition of open sets)

Question

I am reading "Calculus on Manifolds" by Michael Spivak.

In this book, the norm of $x \in \mathbb{R}^n$ is defined by $|x| := \sqrt{(x^1)^2 + \cdots + (x^n)^2}$.
In this book, a set $U \subset \mathbb{R}^n$ is called open if for each $x \in U$ there is an open rectangle $A$ such that $x \in A \subset U$.

Why did Spivak choose this definitions?
What are the advantages of these definitions?

I wonder if we define the norm of $x \in \mathbb{R}^n$ as $|x| := \sqrt{(x^1)^2 + \cdots + (x^n)^2}$, then it is natural we define a set $U \subset \mathbb{R}^n$ is open if for each $x \in U$ there is an open ball $B$ such that $x \in B \subset U$.

And

I wonder if we define the norm of $x \in \mathbb{R}^n$ as $|x| := \max \{|x^1|, \cdots, |x^n|\}$, then it is natural we define a set $U \subset \mathbb{R}^n$ is open if for each $x \in U$ there is an open rectangle $A$ such that $x \in A \subset U$.

Answer 1

Both of them are equivalent. If there is an open rectangle containing the point, there is an open ball within that open rectangle, containing the point. Conversely if there is an open ball containing the point, there is an open rectangle within that open ball, that contains the point.

His definition has the additional advantage that Cartesian product of open rectangles in $\Bbb R^m$ and $\Bbb R^n$ gives us an open rectangle in $\Bbb R^{m+n}$.