Let $X$ be topological space and $A\subset X$. Now, $X$ has a discrete topology.
Determine $\partial A$, the boundary.
Is $\partial A$ empty set?
[Math] Boundary of set A in discrete topological space
general-topology
general-topology
Let $X$ be topological space and $A\subset X$. Now, $X$ has a discrete topology.
Determine $\partial A$, the boundary.
Is $\partial A$ empty set?
Best Answer
Sure. If you know the definition of $\partial A = \overline{A}\setminus A^{\circ}$, then this is clear, as for all subsets $A$ we have $A = A^{\circ}$ (all sets are open) and $A = \overline{A}$ (all sets are closed).
Also, if $A \subset X$ and $x \in X$, the neighbourhood $\{x\}$ of $x$ cannot intersect both $A$ and $X \setminus A$, whatever $A$ is, so $x \notin \partial A$ (this is another equivalent definition of boundary).