Open or closed set

Question

I am stuck on the following problems:

Is the following set open?

$\{x\in\mathbb{R}^3:0<|x|<1, |x|\text{ is irrational}\}$, and

Is the following set both open and closed?

$\{x\in(0,1):x \text{ is rational}\}$

Working:

I am guessing the first one is open because it is a strict inequality, and the second one is open too because it doesn't include 0 and 1.

Is this the right way of thinking? If not what am I missing? Thanks.

Answer 1

You need to provide the construction of $T$.

For the topology in $\mathbb{R}$ introduced by metric, it looks as if both is is not open or closed, because both does not contain all the boundary point and both does not disjoint from the boundary point. ($\mathbb{R}$ is dense, rational can't be separated from irrational by open intervals.)

google:"$p$ is a boundary point of a set if and only if every neighborhood of $p$ contains at least one point in the set and at least one point not in the set. "