Thread: Proof of Topological Property

Show that the boundary of any set D is itself a closed set.

Thanks in advance for any advice...

Originally Posted by jzellt

Show that the boundary of any set D is itself a closed set.

Suppose that $\displaystyle \beta (A)$ is the boundary of $\displaystyle A$.
If $\displaystyle x$is a limit point of $\displaystyle \beta (A)$ and $\displaystyle \mathcal{O}$ is and open set such that $\displaystyle x\in\mathcal{O}$ implies $\displaystyle \left( {\exists y \in \mathcal{O}} \right)\left[ {y \in \beta (A)\backslash \{ x\} } \right]$.
That means that because $\displaystyle y$ is a boundary point, $\displaystyle \mathcal{O} $ must contain a point of $\displaystyle A$ and a point not in $\displaystyle A$.
How does this prove that $\displaystyle x$ is a boundary point of $\displaystyle A$? How does that show that $\displaystyle \beta (A)$ is closed?

$\displaystyle \partial E=\bar{E}\cap\overline{E'}$....sooo...

Maybe state what definition of boundary you are starting from ?