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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.)

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    $\begingroup$ $|0|=0$........ $\endgroup$ – JP McCarthy Sep 12 '17 at 19:59
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    $\begingroup$ Is often called "punctured neighbourhood" as you imagine taking a needle and punching a hole excluding just the middle point. $\endgroup$ – mathreadler Sep 12 '17 at 20:01
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    $\begingroup$ Rule of math: when you hit a wall while learning well known and explored material, check your assumptions, then check your definitions. $\endgroup$ – jpmc26 Sep 13 '17 at 0:14
  • $\begingroup$ Pay very close attention to the inequality. Notice that it uses $<$ and not $\leq$. Ask yourself, how are these two different in this context? $\endgroup$ – Matthew Sep 13 '17 at 4:07
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    $\begingroup$ @DanaHill. I consider them a useless nusiance; perhaps an analysist hang over from the early days of topology. $\endgroup$ – William Elliot Sep 14 '17 at 3:40
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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.

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    $\begingroup$ Okay, so one is "zero is less than or equal to" while the other is "zero is less than". Thanks. $\endgroup$ – Dana Hill Sep 12 '17 at 20:43
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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.

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  • $\begingroup$ Are limit points needed for anything? $\endgroup$ – William Elliot Sep 12 '17 at 20:23
  • $\begingroup$ Sure, we get to limits in chapter 5. :) (we are currently in chapter 3) $\endgroup$ – Dana Hill Sep 12 '17 at 20:42
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    $\begingroup$ @WilliamElliot you're kidding, aren't you? $\endgroup$ – trying Sep 12 '17 at 21:18
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    $\begingroup$ @DanaHill It does not seem so from the high level of accepted answers relative to those you asked. For this particular question do you need further clarification? $\endgroup$ – trying Sep 12 '17 at 23:25
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    $\begingroup$ @trying. No. For topology there is no use except for scattered spaces, a minor topic. So why the big fuss about limit points? What are they use for that is so important that even simple things like closures are awkwardly defined with limit points? $\endgroup$ – William Elliot Sep 13 '17 at 4:02
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If a set $N$ is a neighborhood of a point $p$, then $N-\lbrace p \rbrace$ will be a deleted neighborhood of $p$.

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