In Classical Mathematical Physics (3rd Ed.) (previously A Course in Mathematical Physics), author Walter Thirring cites $M = [a, b]$ as an example of a manifold with a boundary, with atlas $U_1 = [a, b)$, $\Phi_1: x \rightarrow x-a$, $U_2 = (a, b]$, $\Phi_2: x \rightarrow b-x$ ((2.1.21), Example 1, p. 20).
I’m confused, though, because the definition requires that the $\left\{U_i\right\}$ be open sets, whereas both $U_1$ and $U_2$ are only half-open. Other sources suggest that the charts need only be chosen such that the images of the $\left\{U_i\right\}$ are open in $\mathbb{R^n_+}$, but here we have, e.g.,
$$
\Phi_1(U_1) = [0, b-a)
$$
which AFAICT is still only half-open in $\mathbb{R_+}$. What am I missing? And what would be a valid atlas for $M = [a, b]$?
Best Answer
You are missing the concept of relatively open sets. $[0, b) $ is not open in $\mathbb{R}$, but it is an open set in $\mathbb{R}_+$ with its subspace topology (assuming that set defined as $\{x \in \mathbb{R}| x\ge 0\}$)
The definitions from your 'other sources' will work fine with this observation, and with these definitions the atlas you mentioned will also work.
(For subsets $X$ of a topological space $Y$ a set $A\subset X$ is called open in $X$ iff, for every $x\in A$ there is an open set $U\subset Y$ such that $U\cap X\subset A$ and $x\in U$. Note that the set $U$ has to be chosen in the superset $Y$.
If you apply this to the interval $[0,b) \subset \mathbb{R}_+$ you can choose $U = (-1,b)$.)