# Thread: Non-empty, compact, disconnected and limit points

1. ## 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 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

2. ## 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 $|x_k|\leq 2^{-1}\cdot 7^{-k}$.