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March 22

Cluster point & accumulation point

I'm not sure if I should ask this question here or rather at Wikiproject Math's talk page. Feel free to migrate it there, if appropriate.

BACKGOUND

I saw this question How to understand cluster points? at math.stackexchange.com LINK, in which the author seems to confuse the cluster point of a sequence with the accumulation point of a set of values of a sequence's terms. At least, that's how I understand the answer given LINK (the only one given so far), which makes a clear distinction between the two names and their meaning.

I was surprised to find Wikipedia describes both those notions in a single article. What's even more strange, it's named neither of those mentioned, but ...Limit point.

The more I read, the more confused I am. The lead section mentions both notions, one in the first paragraph, the other one at the end. And it says both of them are called with both names mentioned. What a mess! And it's in mathematics, which strives most of all sciences to keep its nomenclature precise and unambiguous.

THE QUESTION(S)

So here is my question: are these names actually commonly used in such an ambiguous way? Isn't any of them prevalent in one of those meanings? Can we do anything about it – that is disambiguate them with separate articles (even if one would be just a little more than a stub)? --CiaPan (talk) 08:30, 22 March 2021 (UTC)[reply]

Just like there are related but distinguishable notions of "limit of a function" and "limit of a sequence", so we can can have related but distinct notions of "accumulation point of a set" and "accumulation point of a sequence".
The definitions given in the Encyclopedia of Mathematics for accumulation point and limit point of a set, although phrased differently, are equivalent. The entry Accumulation point does not mention "limit point", but the entry Limit point of a set states: "A limit point of a set is usually called an accumulation point of that set." There is a rather obscure discussion of terminology at the entry Condensation point: "The three notions [Condensation point of a set; Limit point of a set; Accumulation point] should be clearly distinguished", without explicitly indicating the difference between the last two. Furthermore, we are told, "The term limit point is slightly ambiguous. One might call

x

a limit point of

A

if every neighbourhood of

x

contains infinitely many points of

A

, but this is not standard." However, no enlightenment is provided about what is standard. However, we are told that in the case of a

T 1

-space

X

the notions of a limit point of a set

A \subset X

and an accumulation point of

A

coincide, and that one uses "accumulation point" . The term "cluster point" has no entry in the Encyclopedia of Mathematics, but the entry Limit presents it as a synonym of "accumulation point". What we call an adherent point is called a "proximate point" over there.^[1]
The Handbook of Mathematics by Thierry Vialar writes, "In basic handbooks, accumulation point, limit point, cluster point (of a set) are often regarded as the same."^[2] It defines a cluster point of a sequence as a point, any of whose neighbourhoods contains infinitely many elements of the sequence.^[3]
Neither text defines the accumulation point of a sequence, but a common definition is equivalent to the above definition of "cluster point of a sequence".^{[4][5][6][7][8]} The last cited text has a simple formulation: "A point

x 0

is an

accumulation point

of a sequence

{x q}

if it is a limit of a subsequence of

{x q}

." (The notation is confusing, though; here,

x 0

does not denote the initial element of a sequence indexed starting at

0

, and the braces are not set formers.)
Sometimes the concept of "limit point of a sequence" is defined as being a limit of the sequence,^[9] which I think is confusing. --Lambiam 15:55, 22 March 2021 (UTC)[reply]

(edit conflict) The article already talks about the difference between the limit point of a set and the limit point of a sequence; see the section "For sequences and nets". They are different and they are inconsistent if you ignore the difference between a set a sequence. (The article also mentions nets, which are a generalization.) So yes, you do have to specify "of a set" or "of a sequence" when you say limit point if it's not clear from context, but I don't think it's necessary most of the time; if S is a set you can just say "limit point of S" and it's understood you mean the set S and not some sort of sequence formed from S. I'd argue against disambiguating; the two concepts, though different, are closely related enough that they should be in the same article. It might be clearer if limit points of sequences were least mentioned in the lead since most people will learn about sequences of reals before coming across limit points of set, which are really part of point set topology. Unless things have changed a lot since I was grad student, people generally do elementary real analysis first and general topology a year or two later.

Using the same name or symbol for two different, if related, things is known in programming as Function overloading, for example a+b can be an integer if a and b are integers, and floating point if either a or b is a floating point. Under the hood, integers and floating points are added different ways so they would be implemented with different functions. Overloading seems to be a feature of natural language in general though, including the jargon of every scientific field. This may cause confusion for people just learning a new science, but fortunately we now have sites like SE and Wikipedia to help people. --RDBury (talk) 16:07, 22 March 2021 (UTC)[reply]

This does not help with the answer to your question, but with respect to And it's in mathematics, which strives most of all sciences to keep its nomenclature precise and unambiguous., it is my experience as a mathematician that precision and lack of ambiguity is extremely important locally (i.e., within a single proof, or a single book, words should have clear, precise, unambiguous meanings) but that globally everyone understands that the same words will be used with different meanings, that different notations will be used for the same concept, etc., and that context will take care of the rest. --JBL (talk) 19:31, 23 March 2021 (UTC)[reply]

That is very nicely put. --Trovatore (talk) 19:39, 23 March 2021 (UTC)[reply]

Maybe worth explicitly pointing out the main difference between accumulation points of sets and cluster points of sequences. The different thing about a sequence is that it can have repeated elements, and in particular repeated infinitely often. So consider the sequence 0, 1, 0, 1, 0, 1, ...., which just keeps alternating 0 and 1.
It has two cluster points, 0 and 1, if these are defined as limits of a subsequence. However the set of all points of the sequence is just {0, 1}, and has no accumulation points.
As RDBury points out, in any particular context, you need to determine how the authors are using such terms. Trovatore (talk) 18:41, 24 March 2021 (UTC)[reply]

Stats term

Let's say you're testing how well a marketing scheme is working and you find that in your three trial locations, sales might increase or decrease by varying amounts. Say, one's up by 20%, one's down by 10%, and one's about neutral. In other words, you have variability of about 30 percentage points and are overall up by about 10%. Not bad. But let's say that your control groups (that didn't use the scheme) had different levels of variability: some were up 30%, some down 20%, and so on. Intuitively, I'd know that the marketing data is probably not telling us anything useful because the control groups were all over the place, but what is that situation called? I get that technically it's going to be related to exclusion of the null hypothesis, but I'm curious what the general concept is called. Matt Deres (talk) 15:18, 22 March 2021 (UTC)[reply]

Depending on how the data are processed, this might be called "lies, damned lies and statistics". The whole point of having control groups is to isolate the effect being tested for from other possible effects, such as the waning effect of earlier campaigns. Statistical hygiene requires both the method of data collection and the evaluation method of the date after collection to be decided upon before data collection starts and to remain fixed. If the evaluation method is not fixed in advance, one creates a free-for-all opportunity for cherry picking. In the sketched scenario, any sane test statistic will reveal that the p-value has gone through the roof. Invariably, there are also in(s)ane statistics, such as in this case by ignoring the control groups, that may suggest otherwise. --Lambiam 16:16, 22 March 2021 (UTC)[reply]

Lack of power? BTW, in your example, the intervention group is up 3.33% on average, not 10%. Tigraan^{Click here to contact me} 16:52, 22 March 2021 (UTC)[reply]

Thank you both for the links; I've got some reading to do. :-) @User:Tigraan, it's just a sketch for conversation purposes, but the three test areas could well be different sizes. Matt Deres (talk) 18:05, 22 March 2021 (UTC)[reply]

As a non-statistician, I would call the effect noise, random noise or innate variability.2A00:23C6:AA08:E500:BC6A:ECAD:7310:E8F1 (talk) 19:11, 23 March 2021 (UTC)[reply]