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This question relates to two (seemingly) conflicting definitions of Limit Points in real analysis.

The definition of limit points and closed sets from my notes is written as:

A much more general definition of open and closed sets can be obtained by considering the behaviour of sequences.

Example. Consider the sequence $x_n=\frac{n}{n+1}$. We have $x_n \in (0,1)$, but $x_n\to 1\notin (0,1)$. On the other hand, both the sequence and its limit live in $[0,1]$

Definition: $\text{Consider a set}\, S\subset \mathbb{R}. \text{A point}\,\bar{x}\,$ $\text{is called a limit point of}\,$ $S$ $\,\text{if there exists a sequence}\,$ $x_n\in S\,$ $\text{which converges to}\,\bar{x}$.

Definition: A set $S$ is closed if it contains all its limit points.

and I wish that was all there was to it...

However, I have since seen a different definition from this Wrath of Math video, which clearly states from 0:30 that:

A point $x$ is a limit point of a set $A$ if there is a sequence of points, $a_1,\, a_2,\,a_3,\,\dots$ from $A \setminus\{x\}$ such that $(a_n)\to \,\bar{x}$

So for the second definition, this means that the set $\{1\}$ cannot have $1$ as a limit point, since the trivial $1, 1, 1, \dots$ sequence would satisfy this.

But according to the first definition, this $1$ is actually a limit point (since it does not contradict the requirements given in the quotation).

By way of example, are $\frac{1}{17}$ and $1$ limit points for the interval $\left[\dfrac{1}{17},\,1\right]$?

Well, according to a comment in the chat section below an answer by Randall in this post the answer is yes, $\frac{1}{17}$ and $1$ are in fact limit points.

So which quote is correct?

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    $\begingroup$ Different authors occasionally use different definitions. This is fine; the underlying concepts do not change, they are just presented differently. The former definition also labels elements of the set as limit points. The latter does not. $\endgroup$ Commented 12 hours ago
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    $\begingroup$ Not an answer, but I think it's with mentioning that once you start general topology you'll get a much more general definition of a closed set, and these inconsistencies between different texts won't matter that much. $\endgroup$ Commented 12 hours ago
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    $\begingroup$ @user1540346 In general, I'd say you have to get used to that. Specifically here though, I think the definition of limit point is relatively standard to the point that I'd go so far to say that the definition in your notes is wrong (which doesn't necessarily mean the mathematics developed from such a nonstandard definition is wrong). But note they don't result in different definitions of a closed set (which should be really, really standard). $\endgroup$ Commented 12 hours ago
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    $\begingroup$ I argue that authors should have some freedom in creating definitions that facilitate mathematical facts. This helps at the edge of research, where definitions are not yet settled. In your examples, the authors aim to present that closed sets contain their limit points, and both definitions facilitate it. The facts matter more than the definitions here. $\endgroup$ Commented 12 hours ago
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    $\begingroup$ From a mathematical perspective, there is a better goal for you than insisting on liguistic consistency. I suggest understanding exactly the difference between definition #1 and definition #2: given any $S \subset \mathbb R$, and any $x \in \mathbb R$, understand exactly how to tell when $x$ to satisfies exactly one of those two definitions. This will be a good exercise in preparing for the many, many, many other annoying inconsistencies of terminology that you will encounter in your mathematical career. $\endgroup$ Commented 10 hours ago

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There's no central authority on mathematical definitions, so you may not get an answer of the form “The first definition is ‘wrong’ because...” Authors can make whatever definitions they want, as long as they agree to the consequences.

You're right that there's a conflict between the two definitions you found. What are the consequences? The definition from the ICL notes allows for isolated points to be called limit points, which would preclude having results like:

  • The closure of a set is the disjoint union of its limit points and its isolated points.
  • $x$ is a limit point of $A$ iff $x$ is in the closure of $A\setminus\{x\}$.

If you follow the definition from the video you watched, these are very straightforward to prove.

If you read those ICL notes a bit further, you might notice some strange extra conditions in theorems. You might find a statement about limit points with the additional hypothesis that they aren't isolated. Those theorems would be more cleanly stated and proved with a different definition.

This is how definitions evolve over time. An example I like to point to is prime number. Some will say $1$ ought to be considered prime, since its only positive factors are $1$ and itself (redundantly). But if you let $1$ be a prime number, you give up the uniqueness of factorization into primes. We can tack on as many $1$'s as we like to a factorization. So if you want “unique” factorization, you have to change “prime numbers” to “prime numbers greater than $1$”. Or, you can exclude $1$ from the primes.

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  • $\begingroup$ Aside: what does "ICL" stand for? $\endgroup$ Commented 10 hours ago
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    $\begingroup$ @RobArthan I don't know—I asked OP what notes they were quoting from, and they said “ICL maths dept. on real analysis.” I think it refers to Imperial College of London, but that's only a guess. $\endgroup$ Commented 10 hours ago
  • $\begingroup$ Thanks for the clarification. $\endgroup$ Commented 10 hours ago
  • $\begingroup$ His guess is correct, sorry I was not more explicit. $\endgroup$ Commented 10 hours ago
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Generally, the term limit point or accumulation point is used to refer to the latter definition, so as to distinguish accumulation points from isolated points within the set. The first definition is wrong, or at the very least not preferred, because it cannot filter out isolated points. (Although I would point out that definitions are all conventions, and right/wrong has very little to do with it.)

So the set $\{1\} \cup \left((2,3) \cap \mathbb{Q}\right)$, for instance, would have limit points $[2,3]$. $1$ would not be a limit point.

The former definition is more just a definition of (sequentially) closed sets.

In your example, $1/17$ and $1$ are limit points w.r.t. both definitions since, for instance, we have sequences $1-2^{-n} \to 1$ and $1/17 + 2^{-n} \to 1/17$.

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  • $\begingroup$ Okay, if you say so, still confused though. I guess I will have to think this over. If someone could simply say 'the first definition is wrong because....' Alternatively, 'The second definition is wrong because ....'. That is the sort of explanation I seek. Thank you though! $\endgroup$ Commented 12 hours ago
  • $\begingroup$ @user1540346 Did an edit to make things clearer, I hope. $\endgroup$ Commented 12 hours ago
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    $\begingroup$ @user1540346 One can not say a definition is wrong since it is not a statement. So we can say a theorem is wrong but can not say a definition is. The second definition is more widely used, but if for the first definition (say it is from book A), all following theorems in book A involving limit points are correct if limit points are understood as the first definition, then this would be fine. $\endgroup$ Commented 12 hours ago
  • $\begingroup$ Also, if for whatever reason the former author wants you to consider the set of limit points of the latter author, then they might ask you to consider $\bar S-S$. $\endgroup$ Commented 11 hours ago
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Intuitively there are points that are "right up next to a set". Those points are important.

We formally define $x$ is "right up next to the set" $S$ if: For every open neighborhood around $x$, the neighborhood contains an element of $S$.

This means that if $S$ is the set $(1,3)$, say, the point $1$ is "right up next to" $S$ because every neighborhood of $1$ will "jut into" the set $(1,3)$. A neighborhood with radius "$1$ micron" will contain the point "$1+\frac 12$ a micron that is in $(1,3)$.

(And obviously the point $02.997$ is not "right up next to the set" $(1,3)$ because we can take a really small neighborhood with radius $0.0029$ (so the neighborhood is $(0.9941, .9999)$ that misses $(1,3)$ entirely).

But what if our set is $\mathbb Z$? Obviously the non-integers don't lie "right up next to the set" $\mathbb Z$. But what about the integers themselves.

Is $3$ a limit point of $\mathbb Z$?

If you take really really small neighborhoods of $3$ the only integer that is in that neighborhood is $3$ and every neighborhood around $3$ will have to contain $3$ (as, duh, $3$ is the very center of the neighborhood).

Does this count as but "right up next to the integers"? On the one hand it seems like it must. $3$ is an integer and you can't get any "closer" or right up "next to" than actually being in the set. But on the other hand this seems "unfair" and missing the point. Surely by "lying up right next to" the implication is the point $3$ is "next to" something else other than itself.

So there you have it: Definition 1: is saying points themselves are "right up next to themselves" and a limit point is any point, whether a member of the set or not.

Definition 2: is saying points can't be "right up next to themselves" and "right up" implies being "next to" something elss. So a limit point is any point (whether in the set or not) that is "right up next to the set" in that every neighborhood of the limit point contains a member of the set (that differs from the point itself).

So which is "correct"?

Does it matter?

(Okay, confession time. I am very much in the "The second definition is correct" camp and I really do not like definition one. But...)

It doesn't really matter.

The main idea is that we want to define a "closed" set as being a set that contains all these limit points that lie up "right next to it". It doesn't matter if these isolated points that are only next to themselves and no other count or not. Because they are elements of the set they are going to be included as points in the set whether we think they lie next to themselves or not. In defining "closedness" we are concerned whether there are any points not in the set but are nonetheless "right up next" to it. If there are, the set is not closed. But if all points "right up next" are in the set then the set is closed. There's no question about these isolated points being in the set, so whether we think of them as isolated and not "next to" the set or whether we think of them as being "next to themselves" won't matter.

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It so happens that the most widely accepted current convention is to define "limit point" or "accumulation point" as per the second cited definition in your question, which is indeed confusing that it is not equivalent to just being a limit of some (possibly constant) sequence. So it may be good to avoid using the term "limit point" in favour of "accumulation point".

The other notion is called "adherent point". One can guess that your course instructor wants the term to invoke the idea "limit of points", but as you can see it is not the standard definition.

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