# Interior of a Set

1. Feb 12, 2010

### Piglet1024

I have to describe the interior of the subsets of R: Z,Q.

I don't understand how to tell if these certain subsets are open or how to tell what the interior is, can someone please explain

2. Feb 12, 2010

### owlpride

Are you working with the standard open-ball topology on the real line?

If so, then a point $p \in S \subset \mathbb{R}$ is an interior point of S if for some $\epsilon >0$, the open interval $(p-\epsilon, p+\epsilon)$ lies completely inside. In other words, a point is an interior point if it lies in the set and is not a boundary point of the set.

For example, 2 is an interior point of [1,4], but 1 is not an interior point (on the boundary) and neither is 0 (not in the set).

What happens when you draw a small open interval around a rational number? Will that interval lie completely inside the rational numbers, or does it contain an irrational number?