What's a "deleted neighborhood"? (other than very very confusing)

My text has the following definitions:

3.4.1 DEFINITION Let $x\in\mathbb{R}$ and let $\epsilon>0$. A neighborhood of $x$ (or an $\epsilon$-neighborhood of $x$)† is a set of the form $$N(x; \epsilon) = \{y\in\mathbb{R} : |x-y|<\epsilon\}.$$

3.4.2 DEFINITION Let $x\in\mathbb{R}$ and let $\epsilon>0$. A deleted neighborhood of $x$ is a set of the form $$N^*(x; \epsilon) = \{y\in\mathbb{R} : 0 < |x-y| < \epsilon\}.^‡$$

Please explain to me how these are different. The ONLY change I see is the second one has $0<$, which I don't see as necessary as the absolute value is ALWAYS positive. It's part of its definition.

I've tried getting clarification from my professor and the TA, and it's just SO confusing. (Mostly this comes from trying to put accumulation points into context, as its definition comes from the deleted neighborhoods.)

(Definitions from Analysis, With An Introduction to Proof, by Steven Lay. Page 135.)

Solution 1:

The first contains $x$ (as the distance is allowed to be $0$) while the second excludes $x$.

Also, the absolute value is NOT always positive. It is always non-negative. $0$ matters.

Solution 2:

It is also called punctured. It's the neighborhood of a point minus the point itself. It's very useful as a tool to know whether a point is a limit point of a set. A point is a limit point of a set when any deleted neighborhood of such a point has a non-empty intersection with the set.