I am trying to teach myself some basic tensor analysis and differential geometry. Right now, I am reading Abraham, Marsden, and Ratiu's Manifolds, Tensor Analysis, and Applications. In the first few sections, they define the boundary of a subset $A$ of a topological space $S$ as the intersection of the closure of $A$ with the closure of $S/A$. I was going through examples of this in my head and have run into a conceptual problem, possibly due to a misunderstanding. Perhaps someone can help?
Let $S$ be the usual topology of open sets on $\mathbb{R}^3$ and let $A$ be the unit sphere. I want to find $bd(A)$. It is easy to see that $cl(A) = A$, but I do not understand how to find $cl(S/A)$. My first guess is that it is empty, since I cannot think of any closed sets which contain $S/A$ as a proper subset. But then that implies
$bd(A)=cl(A) \cap cl(S/A)=A\cap\emptyset=\emptyset$
which sort of goes against my intuition. Am I missing something here, or is this a true statement?
You're correct that $A$ is closed. But the closure of $(\mathbb{R}^3\setminus A)$ is $\mathbb{R}^3$. (In any topological space $X$, $X$ is a closed set. The only set $Y$ with $\text{cl}(Y) = \emptyset$ is $Y = \emptyset$.)
So $\text{bd}(A) = A\cap \mathbb{R}^3 = A$.