# Non-empty, compact, disconnected and limit points

• March 7th 2012, 05:48 AM
klw289
Non-empty, compact, disconnected and limit points
I am at the moment trying to get through some basic set theory and I'm getting very stuck with the proofs. This question is from a textbook I am studying from and as it is a prove question there is no solution in the back :(
Any help with explanations would be very useful

Let Attachment 23327 where E0 = [0, 1] and Ek is constructed by removing the
second, fourth and sixth sevenths of each component Ek-1. Each set that is removed is open.

I need to show that F is non-empy, compact, totally disconnected and that 0 is a limit point of F
• March 7th 2012, 06:23 AM
girdav
Re: Non-empty, compact, disconnected and limit points
Show that each $E_k$ is compact, which will give you the first two points. For totally disconnectedness, show that $E$ doesn't contain any interval. Show that for each $k$ there is $x_k$ such that $|x_k|\leq 2^{-1}\cdot 7^{-k}$.