Can a single point be called a region? I'm reading $\delta$-neighborhood in my calculus book, Calculus, Larson, 10th ed. p.880, and it's not clear for me what's the precise definition of region:
If a single point could be a region, then that point would not be a interior point. And it's not clear whether points inside $R$ including that point. If it is not included then it is not a boundary... So wt_ is that point?
If $R$ is a subset of $ \mathbb R^2$, then $R$ is called open if every point of $R$ is an interior point of $R$.
$R$ is called a region if $R$ is open and connected and $R \ne \emptyset$.
See: https://en.wikipedia.org/wiki/Region_(mathematics)
So I can not agree with the definition in your book