What does $\partial A$ mean? I'm having trouble understanding what $\partial S$ means in analysis?
Let S be a subset of $\mathbb{R}$.  Show that $\partial$ S is closed;
Just curious what the partial symbol means in this case.
 A: Most likely $\partial A$ denotes the boundary of $A$; that is, the closure of $A$ minus the interior of $A$. For example, the boundary of the interval $(0,1]$ is $[0,1]\setminus(0,1)=\{0\}\cup\{1\}$.
A: The topological boundary $\partial S$ of a subset $S$ of a topological space $X$ can be defined as the intersection of the closure and the complement of the interior, i.e.
$$
\partial S := \overline{S} \cap (S^\circ)^C
$$
since both are closed so is their intersection.
A: The $\partial S$ is usually denote by the boundary of a set in analysis, and in analysis usually talk $\partial S$ about some subset of $R^n$, $\partial S$ means the set which element have the property that any neighbourhood of the element have some point of $S$ and the some point of the complement set of $S$.
when $S$ is a subset of $R$, The cluster point set of $\partial S$,denoted by $A$, any element of $A$,any neighbourhood of the element contain a element of $\partial S$, then contain a neighbourhood of the element of $\partial S$,so contain  some point of $S$ and the some point of the complement set of $S$,so belongs to $\partial S$,so it's closed.
