# Proof of Topological Property

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

• February 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 $\beta (A)$ is the boundary of $A$.
If $x$is a limit point of $\beta (A)$ and $\mathcal{O}$ is and open set such that $x\in\mathcal{O}$ implies $\left( {\exists y \in \mathcal{O}} \right)\left[ {y \in \beta (A)\backslash \{ x\} } \right]$.
That means that because $y$ is a boundary point, $\mathcal{O}$ must contain a point of $A$ and a point not in $A$.
How does this prove that $x$ is a boundary point of $A$? How does that show that $\beta (A)$ is closed?
• February 3rd 2010, 01:23 PM
Drexel28
$\partial E=\bar{E}\cap\overline{E'}$....sooo...
• February 3rd 2010, 02:18 PM
mabruka
Maybe state what definition of boundary you are starting from ?