- #1

- 67

- 5

## Homework Statement

The

*boundary*[tex]\partial E [/tex] of a set E if defined to be the set f points adherent to both E and the complement of E,

[tex] \partial E=\overline{E}\bigcap \overline{(X\backslash E)}[/tex]

Show that E is open if and only if [tex] E \bigcap \partial E [/tex] is empty.

**Show that E is closed if and only if [tex]\partial E \subseteq E [/tex]**

I did the first part, but I need help with the second part.

## Homework Equations

## The Attempt at a Solution

Assume E is closed, then [tex] E = \overline{E}[/tex] and its complement is open

so, [tex] (X\backslash E) \subset \overline{(X\backslash E)}[/tex] and [tex]\overline{(X\backslash E)} [/tex]

(contains points in X but not in X\E)

So, [tex]E\bigcap \overline{(X\backslash E)}=\overline{E} \bigcap \overline{(X\backslash E)}[/tex] is non empty and every point in [tex]\overline{E} \bigcap \overline{(X\backslash E)} [/tex] is in E since [tex]E=\overline{E} [/tex]

So, [tex]\partial E \subseteq E[/tex]

But I am having trouble going in the other direction

Last edited: