I am reading "All the math you missed" by Thomas A. Garrity, second edition. In the chapter on Divergence Theorem there is the following fragment:
Here the derivative $\frac{df}{dx}$ is integrated over the interval $$[a, b] = \{x \in \mathbb{R}: a\leq x \leq b\},$$ which has as its boundary the points a and b. The orientation on the boundary will be b and -a, or $$\partial [a, b]=b-a.$$
I am confused by the fact that $\partial$ was defined previously as boundary of manifold, which is a set, while $b-a$ is clearly not a set. Is this notation overloaded (possibly implicitly) and varies depending on the context or am I simply misunderstanding something? Or perhaps there is typo in the book and the statement should be different? I have seen some lecture notes that used symbols like $M$ for manifold, $\imath$ for its orientation and $\partial\imath$ for induced orientation of $\partial M$, but this notation is still not the same as the one I am struggling to comprehend.
The boundary of an interval is indeed a set of 2 oriented points, let's write it as $\partial [a, b] = \left\{a_-, b_+ \right\} $. "Integrating" a function $f$ over this set should be just evaluating it on the 2 points times the corresponding orientation, so $f(b)-f(a)$