is there any reason for why we are restricting ourselves to the only the upper half-space of $\mathbb{R}^k$ in Munkres' book ?

Question

In the book of Analysis on Manifolds by Munkres, at page 200, it is given that

However, is there any reason for why we are restricting ourselves to the only the upper half-space of $\mathbb{R}^k$ ?

Answer 1

You can take any affine hyperplane $H \subset \mathbb{R}^k$ (see https://en.wikipedia.org/wiki/Hyperplane). $H$ is determined by two parameters $a \in \mathbb{R}^k$, $a \ne 0$, and $b \in \mathbb{R}$ as the set of all $x$ such that $a \cdot x = b$, where $a \cdot x$ denotes the usual dot product.

$H$ determines two closed halfspaces as the set of all $x$ such that $a \cdot x \ge b$ and $a \cdot x \le b$, respectively. The open halfspaces are defined similarly (via > and <).

For any two closed halfspaces there exists an affine isomorphism of $\mathbb{R}^k$ mapping one to the other. This means that all closed halfspace have "equal rights".

$\mathbb{H}^k$ is given by $b = 0$ and $a = (0,..., 0,1)$ with $\ge$. This is just a nice and customary choice but does not have a compelling reason.