# Open subset of a perfect set.

1. Sep 18, 2013

### kostas230

Suppose we have a perfect set $E\subset\mathbb{R}^k$. Is there an open set $I\subset E$?

2. Sep 18, 2013

### mathman

I'm rusty at this. However, I understand that a closed interval is a perfect set. Take the closed unit cube in Rk, drop all boundary points leaving an open unit cube.

3. Sep 18, 2013

### Office_Shredder

Staff Emeritus
Sometime's it's true (like mathman's example), and sometimes it's false. For example the Cantor set is a perfect set but contains no open interval inside of it

4. Sep 19, 2013

### kostas230

Yeah, I just found out that. Thank you :)