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.)

If a set $N$ is a neighborhood of a point $p$, then $N-\lbrace p \rbrace$ will be a deleted neighborhood of $p$. 
