# Talk:Limit point

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## Changes

You seem to be in the middle of a list of changes to this page. I won't interrupt, other than to point out:

• the version of the page without the TeX code formats better. The [itex] stuff should be as a last resort, from what I've read.
• your last paragraph is incorrect, in that open sets usually do contain limit points. For instance, the set of limit points of the open interval (0,1) is the closed interval [0,1], not just the endpoints. Some qualification is necessary - e.g. limit points other those in the set.

Cheers, AndrewKepert 03:14, 13 Nov 2003 (UTC)

Okay, since Dcoetzee's changes have finished, I think almost all the changes should be reverted back to the 7 Jun 2003 version. The last paragraph is factually incorrect. Okay if you say "boundary point" instead of "limit point" but otherwise no. So I am removing it - sorry. The use of math code instead of html character entities is nicer in some ways but really makes the document less readable, mainly due to it pinning the font size to a particular pixel size, and hence resolution dependent. The rule of thumb is that if it can be done without [itex] then this is preferable.
The two improvements are the speling corection (containg -> containing) and the change of "infinitely many" to "at least one other".
So my action is to revert to the 7 Jun version and incorporate these changes. Someone else may want to put in a corrected version of the last paragraph.
AndrewKepert 06:58, 13 Nov 2003 (UTC)
I'm sorry, you're right, I wasn't thinking straight. The part about the closure being the union of a set and its limit points is true, but the rest wasn't. Sorry for the bad changes.
I'm still a little new to Wikipedia, I just figured out talk pages. The HTML codes you use appear in my browser (IE) as plaintext, so I can't tell what the article is supposed to be saying. I think the font problems associated with the TeX markup are better than this. This is a subject of some debate on the page
http://meta.wikipedia.org/wiki/Math_Markup
Derrick Coetzee 03:23, 15 Nov 2003 (UTC)

## Proposition

I added a comment to the definition, I hope this is okay. Also, Proposition 1 is not sharp; it holds even if the space is not T0. It would probably be better to substitute a proof which does not use it (I know of at least one simple such proof).

## HTML codes versus math tags

Because most viewers use IE, and it does not support the HTML tags originally used, I believe it's preferable to use the math tags. Although inline images can be unpleasant, some of the symbols translate into characters in the Symbol font, which IE will render, and these are just as good. Also, the user can choose how to render math tag symbols in their preferences, and in the future when browsers support MathML and Wikipedia translates these tags to MathML, it will be able to render even the more complex symbols without sending inline images. I hope this is sufficient justification for you. View the web page I linked above for more discussion of this.

Oh okay - fair enough. I use Mozilla, and the symbols were listed on Wikipedia:How to edit a page and looked ok to me. So I assumed they were widely implemented in common browsers. Maybe the wiki engine should re-encode these on the fly if (a) user prefs dictate or (b) it detects IE? (via [itex] or symbol fonts) AndrewKepert 01:36, 20 Nov 2003 (UTC)

I had written about an hour's worth of explanation and comments, but since I forgot to cut and paste, and lost contact with wikipedia (not the net), it's gone forever, so I'm not even going to try to rewrite it, I don't have time. Be warned, though: IF YOU USE INTERNET EXPLORER, ALWAYS CUT AND PASTE TO A NOTEPAD OR WORD SOMETHING YOU'RE GOING TO SEND TO WIKIPEDIA BEFORE YOU FINALLY SEND IT. IF YOU DON'T, YOU COULD LOSE IT FOREVER. So, you might think the following comments are terse and/or rude, I'm sorry, blame Bill Gates or the idiots who wrote Explorer or the wiki servers or something.

First, there are 3 distinct definitions, whether you count the point, look for a distinct point, or look for infinitely many points. By "limit point", most people mean a distinct point. The first is really "point in the closure". The second is NOT the same! So, I suggest that limit point be used for what it is currently, and "accumulation point" for infinitely many (which is not the same as simply distinct -- consider Sierpinski space I believe). "Cluster point" is another matter, depending on what opinion people who use filters have for terminology.

[edit out]

My bad! I must really be fuming at these computers. The reformulation is correct...but the Sierpinski space is a counterexample for LP vs. AP, I think. Revolver

I looked up in some old textbooks and found:

• "limit point" is hardly ever used for "point of closure", sometimes this is called "point of adhesion" or something similar, so I'll take that part out.
• T1 is equivalent to every limit point being accumulation point -- proof to come. (It's an interesting problem)

I'll clean up a bit, add proof later today... Revolver

I like the discussion of the three different definitions and terms, but there's enough consistent usage of all three for definition (2) in particular that I think it's a mistake to define accumulation point to refer to (3). I changed this.
Derrick Coetzee 14:10, 3 Dec 2003 (UTC)

I still think accumulation point should be defined to mean (3), and I'll explain. The problem is that by defining limit point and accumulation point to mean the same thing, then all the articles and examples that involve non-T1 spaces immediately lose a piece of terminology to lose. Put another way, Wikipedia terminology (at least in mathematics) has to appease even the slightest minorities. Yes, for 80-90% (or more) of the people (or articles) there's consistent usage, but for 10-20% of the people or articles, it's not consistent enough, in the sense that the distinction must be made. Put another way, "consistent usage" is a very subjective judgement, esp. when made by people in mathematics who may work in differing areas of research or work on different problems. So, I think this outweighs the confusion that comes from defining them both to mean what we mean by limit point. To give an example, I could write articles on Sierpinski space or the upper- or lower-limit topologies on the reals, or other articles, and in all of these, I might at some time want to talk about "limit points" and "accumulation points" as separate definitions. To define them to be the same takes this possibility away. At the same time, if an article only deals with T1 spaces, then the authors will be able to use both "limit point" and "accumulation point" freely, because in those cases, it won't make a difference. In other words, if articles have been using "accumulation point" when they really meant "limit point", there's no need to go back and change everything, or even to mention the distinction, so long as the articles in question are only discussing T1 spaces. Now, if the articles ARE discussing non-T1 spaces, then there IS a need to go back and possibly cause confusion, because the article is attempting to tackle examples where the two definitions aren't the same. So, even by defining them differently, it shouldn't be confusing for those articles where they're equivalent. Revolver

Sorry...I keep changing things back, to explain the alternate definition..."neighbourhood" def is not the same as "open set" def, not every neighbourhood is open (see wikipedia definition of neighbourhood, contains an open set), so there is technically a slight distinction between the two definitions. Revolver

In reference to this particular point, I have seen neighbourhood used in this sense. I would use the term open neighbourhood.
Derrick Coetzee 23:54, 4 Dec 2003 (UTC)
Oops, I see what you mean now. You're actually making a stronger statement.
Derrick Coetzee 23:57, 4 Dec 2003 (UTC)

Okay, I've been thinking about this more. I went back and looked at several standard textbooks, and most (not all) do define accumulation point the same as we've defined limit point. So it might be confusing not to do this. At the same time, I think there's a need for some terminology to refer to what I want to call "accumulation point". Perhaps a compromise (or the solution that would be most comfortable to the most articles) would be to let accumulation point be the same as limit point, but have another term for "accumulation point", a standard one seems to be "omega-accumulation point" (I assume omega refers to the ordinal number, which is strange since it's a matter of cardinality, not ordinality). Then, maybe this whole discussion of the distinction could be moved to a separate article on the new definition. How does this sound? Revolver

Sorry if I seem a bit touchy above. General (point-set, descriptive) topology is a sort of guilty pleasure hobby for me, so I've explored more than a bit "life without T2" or some of the bizarre spaces one can get from considering various cardinalities or even different axiom systems. But I'm also quite rusty on some of the terminology. Let me check out some stuff at the library and see if there's a consistent terminology for some of this stuff. I know there is some terminology that distinguishes say nbhds of a point that meet a set in a distinct, infinitely many, uncountably many points, e.g. the "uncountably many" concept is sometimes called a "condensation point", although presumably one could define the concept for any cardinal at all. Revolver

## Trivial topology statement

If a space has more than one point, then its topology is trivial if and only if every point is a limit point of every nonempty subset.

I'm afraid this simply isn't true; we already pondered this for a while in the Trivial topology article, and our conclusion was instead this:

If S is any subset of X with more than one element, then all elements of X are limit points of S. If S is a singleton, then every point of X \ S is still a limit point of S.

This is easy to see if you think about the closure-based definition of limit point. I'm replacing the statement and proof.

Derrick Coetzee 13:33, 5 Dec 2003 (UTC)

## Ambiguity

"infinitely many points $\displaystyle x_{n}$ such that ${\displaystyle d(x,x_{n})<\epsilon }$" is ambiguous when there may be infinitely many n for which $\displaystyle x_{n}$ are equal. Probably this should be "infinitely many values of n such that ${\displaystyle d(x,x_{n})<\epsilon }$.--Patrick 23:22, 2 June 2007 (UTC)

I fixed this. Since, if for infinitely many n the values of $\displaystyle x_{n}$ are equal, this point is a cluster point of the sequence but not necessarily a limit point of the set of points in the sequence, care should be taken with terminology so that the two are not confused. I edited the article accordingly. Any comments?--Patrick 12:30, 6 June 2007 (UTC)

## Differences between limit point, accumulation point, and cluster point?

According to PlanetMath and MathWorld they are the same. Others sources including this one disagree. What is correct? Ht686rg90 (talk) 16:30, 27 June 2008 (UTC)

I think these notions are the same. Oleg Alexandrov (talk) 02:54, 28 June 2008 (UTC)
Agreed. By introducing sequences, the article section in question manages to confuse the concept of "limit point of a set" with the separate concept of "limit of a sequence". So the set {0.1, 0.9, 0.01, 0.99, 0.001, 0.999 ...} has two limit points/cluster points/accumulation points at 0 and 1, but the sequence 0.1, 0.9, 0.01, 0.99, 0.001, 0.999 ... has no limit.
I think the confusion was introduced in a set of edits in June 2007 by User:Patrick, which he mentions in the talk page section above. I suggest we remove all mention of sequence limits from this article and go back to something close to this May 2007 version of the article. Gandalf61 (talk) 08:59, 28 June 2008 (UTC)
Done. This is not the first time I see Patrick doing odd edits in math articles. Oleg Alexandrov (talk) 16:00, 28 June 2008 (UTC)

## Characterization of limit point in sequential space

"Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence of points in S whose limit is x; hence, x is called a limit point."

I think the "only if" part is wrong. It is true for "point of closure" (also called "adherent point"), but not for isolated points. E.g., in R, let S be the union of [0,1] and {5}. Then {5} is an isolated point; it is not a limit point (in the definition of "every nhbd meets a point in S different from it"), but is a point of closure. But obviously the constant sequence equal to 5 is a sequence in S, which converges to 5.

What is true (and useful) is replacing S by S-{x}, i.e. requiring the sequence to consist of elements different from x itself:

"Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence of points in S-{x} whose limit is x; hence, x is called a limit point."

Urysohn (talk) 20:03, 17 June 2010 (UTC)

Yes, that's a silly omission. Fixed.—Emil J. 11:10, 18 June 2010 (UTC)

## Point Of Accumulation KEY CONCEPT

"Every finite interval or bounded interval that contains an infinite number of points must have at least one point of accumulation."

I added this to the bottom of the opening paragraph and it was quickly deleted. This is the key concept. Please see Richard Courant.

This article is actually very poorly written. It does not include this key concept nor does it expand on it. This concept is fundamental to calculus. 01001 (talk) 18:42, 10 March 2011 (UTC)

## Limit point of sets with cardinality 1

"[...] every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself."

Having S:={x} consisting of one element only and, for instance, the discrete topology on X, x is the limit of any sequence on S, respectively ALL elements of S are apparently in a neighbourhood of x, thus x should be "intuitively" a limit point. However, it fails the condition above. I think the definition is not well chosen and should include that case. --85.0.115.52 (talk) 16:46, 1 January 2014 (UTC)

Ok, I see, the definition is chosen intentionally, as "No isolated point is a limit point of any set." --85.0.115.52 (talk) 17:01, 1 January 2014 (UTC)