Can a Perfect Set Contain an Open Subset?

[kostas230]

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

 

[mathman]

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

 

[Office_Shredder]

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

 

[kostas230]

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