# Proof of Topological Property

• Feb 1st 2010, 09:29 PM
jzellt
Proof of Topological Property
Show that the boundary of any set D is itself a closed set.

• Feb 2nd 2010, 06:31 AM
Plato
Quote:

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?
• Feb 3rd 2010, 01:23 PM
Drexel28
$\displaystyle \partial E=\bar{E}\cap\overline{E'}$....sooo...
• Feb 3rd 2010, 02:18 PM
mabruka
Maybe state what definition of boundary you are starting from ?