What is the role of the curly d operator in defining topological boundaries?

[Gauss M.D.]

Our lecturer hastily referred to the boundary of a set M as ∂M, then he dropped it. It sounded very interesting, but he said it was outside the scope of the course. We have also been told that the curly d:s do not allow manipulation in the same way as regular differentials. But given something like ∂z/∂x = 2x + y, you're allowed to move the ∂x to the right side and integrate. So what gives? What exactly does the curly d operator do to a function?

 

[WannabeNewton]

Let $X$ be a topological space and $A\subseteq X$. We define the closure $\bar{A}$ of $A$ as the smallest closed subset of $X$ containing $A$ i.e. $\bar{A} = \bigcap \left \{ B\subseteq X:B\supseteq A \text{ and B is closed in X} \right \}$. Similarly we define the interior $\text{int}A$ as the largest open subset of $X$ contained in $A$ i.e. $\text{int}A = \bigcup \left \{ C\subseteq X:C\subseteq A \text{ and C is open in X} \right \}$. Using these two topological notions, we define the boundary $\partial A$ of $A$ as $\partial A = X\setminus (\text{intA}\cup (X\setminus \bar{A}))$. Intuitively, it is the part "in between" the exterior and interior of $A$. For example, the boundary $\partial A$ in $\mathbb{R}$ of the set $A = (0,1)\subset \mathbb{R}$ is just $\left \{ 0,1 \right \}$, as would be expected.

There is a very nice theorem relating paths in topological spaces to boundaries of sets which makes the concept of the boundary even more geometric but as your professor said these concepts might very well be beyond the scope of your course.