Talk:Derived set (mathematics)

Why does perfect set redirect here?

A perfect set is not a synonym for a derived set, nor is perfect set closely tied to the terminology derived set. A perfect set is a closed set with no isolated points. The fact that you can express this in terms of derived sets is irrelevant.

Google "perfect set" you get the definition for derived set because of this idiotic redirect. Whoever is doing such things, please stop making Wikipedia unusable by deleting useful pages. I'm recreating the perfect set page. — Preceding unsigned comment added by 96.21.160.133 (talk) 18:00, 29 March 2013 (UTC)

S** subset S*

This is not a property of the derived set operation: Consider the set {x,y} with the topology where only the whole set and the empty set are open. Take S={x}. Then S'={y} and (S')'={x}

I don't know how old this unsigned remark is, but: If S={x}, then S*={x,y} and not only {y}. ! Therefore we have that S** is a subset of S*, since S** and S* are actually equal (in this example). --131.234.106.197 (talk) 15:32, 19 December 2007 (UTC)

We say that a point x in X is a limit point of S if every open set containing x also contains a point of S other than x itself (Wikipedia). Hence, in the example above, x is not a limit point of S={x}: there is an open set {x,y} containing x which does not contain a point from S={x} other than x itself. —Preceding unsigned comment added by 146.50.15.89 (talk) 15:07, 5 August 2008 (UTC)

• Hmm. That example also violates condition 3' (but not condition 3, which is claimed to be equivalent to it, but it looks like isn't). In fact conditions 1 and 3' make the derived set of any finite set empty. On the other hand, I think 1,2,3',4' are correct for T_1 spaces. --Unzerlegbarkeit (talk) 17:10, 13 August 2008 (UTC)

Relation to closure

What's the relation between the derived set of a set, and the closure of that set? It would seem that in metric spaces the concepts coincide, no? -GTBacchus^(talk) 07:28, 4 December 2006 (UTC)

A derived set might not contain some of the points of the original set, eg. isolated points. But the closure of a set is the union of the set with its derived set. 192.75.48.150 17:41, 8 January 2007 (UTC)

Bendixson

Bendixson proved that every uncountable closed set can be partitioned into a perfect set, called the Bendixson derivative of the original set and a countable set.

The name "Cantor-Bendixson theorem" rings a bell, and I think it was something like this, but this isn't quite right is it? Take an uncountable set which is totally discrete. It is an uncountable closed set, as required. But the only perfect set is empty, and the remainder is uncountable. Maybe there's some condition missing here. Google doesn't help and I don't have access to my dead trees at the moment. 192.75.48.150 17:32, 8 January 2007 (UTC)

Thanks for pointing that out. I fixed the statement to make it correct. I don't know if it really belongs here, however. CMummert 17:43, 8 January 2007 (UTC)

Streamlining the article a bit

@Wcherowi: Hi, I see from the history section that you have already edited quite a few things lately on that page, so I don't want to step on any toes. I had removed the definition of T1 space in order to tighten the article a bit, to make it more to the point, which is to discuss "derived set". My thinking is that most readers of this article will already know what a T1 space is, and if they don't they can just click on the link. For people who know what T1 means, the redefinition is annoying and dilutes the value of the article. There is no need to redefine every single notion that is used in a article if the notion itself is defined independently of the concept being discussed. That's what links are for (I think that's a general guideline of wikipedia).

FYI, for an extreme violation of that principle see Appert topology, which in this old revision https://en.wikipedia.org/w/index.php?title=Appert_topology&oldid=868847141 used to repeat the full definition of what a topology is, something completely ridiculous and confusing in that particular context.

What do you think? PatrickR2 (talk) 05:57, 29 August 2020 (UTC)

Wikipedia is constantly being criticized for having math articles that only mathematicians can read. I think that this criticism has some validity to it. After all it is mathematicians who are writing these articles and we have spent lifetimes writing for mathematical audiences. While conciseness is a virtue in mathematical writing, it is not one in encyclopedic writing. The audience has changed and we have an obligation to try to make our articles as accessible as possible. There has been a discussion among members of the math project (although it has been several years ago) of "definition and link" vs. "link alone". I don't think that there was any consensus on the matter and there were definitely some that held your view, but I came away from that discussion with the feeling that short definitions with a link was the way to make our pages more accessible. I am not saying that all basic definitions need to be repeated, but short ones certainly reduce the number of clicks one needs to read a dense article. You say that those who know what a T₁ space is would be annoyed by the brief definition, but I claim that those who do know what it is would probably not be interested in this page. Also, those of us who are not topologists, would appreciate a quick recall of a concept rather than having to chase it down in another article (which in this case is buried in a list of properties).

As to the old version of the Appert topology article, I prefer it. The lead of an article is meant to entice the reader to read more. As there are several ways to define a topology, presenting one that is used in the rest of the article makes a lot of sense. A reader who is not familiar with topological concepts, when faced with terms that come out of left field will quickly leave the page, defeating the purpose of having an encyclopedia.--Bill Cherowitzo (talk) 20:52, 29 August 2020 (UTC)

I don't want to discuss the Appert topology article at too much length here. But the point is that each article lives within a certain context. The Appert topology example's context is Counterexamples in Topology, a well known book providing a compilation of examples and counterexamples in topology, and that page references multiple other pages showing those examples in more detail. Many of the examples there are advanced and not meant to be digested by a beginner. A more introductory article on what a topological space means can certainly expand at length on any basic topic as appropriate, but there is no need to repeat the basic general definition of what a topology is in every single article presenting the details of a particular example of a topological space. (and actually Appert topology was the only one I saw that was repeating the general def of a topology). A beginner that first gets acquainted with the general notion of a topology in some of the more general pages should not have difficulties progressing to pages requiring more background once they get more experience. Each article is assuming a particular level of understanding from its readers, and as readers progress in their understanding, they become better able to grasp more advanced articles.

WP:NOTTEXTBOOK is supposed to be one of guiding principles of Wikipedia: "Wikipedia is an encyclopedic reference, not a textbook. The purpose of Wikipedia is to present facts, not to teach subject matter." In the context of mathematics, that can be open for discussion. I myself appreciate some amount of explanations for many articles.

But to take your point of view to an extreme, we freely use the words "set" and "subset" in many articles for example. Why not redefine in each article what "set" and "subset" mean? It's certainly appropriate to do so in some articles, but I think most of us would agree that it would negatively impact the quality of most other articles that just take the notions of set/subset as known prerequisites, by burying the useful focused information of that article in a morass of irrelevant peripheral definitions ("why is the author defining 'subset' here? Is there a hidden meaning to this? Is the notion being used in a non-standard way? etc")

That being said, we can agree to disagree. I won't fight for the deletion of the redefinition of T1 space here, it's no big deal in this case. Regards. PatrickR2 (talk) 23:25, 29 August 2020 (UTC)

Removing irrelevant paragraph

@Mgkrupa: I am proposing to remove the paragraph mentioning separated sets from this article. It is true that separated sets can be described in terms of derived sets (or equivalently in terms of closures). I am not objecting the validity of this fact in any way. My point is that it does not add any information to the notion of derived set, and if someone really cares about this fact (even though the equivalence between the two characterizations of separated sets is really a triviality), then it should be added to the article about separated sets and not here.

To give an analogy, the notion of vector space is ubiquitous in mathematics. If certain mathematical concept XXX can be described in terms of vector spaces, there can be a link from the XXX article to vector space. But adding to the vector space article a link to the XXX article just because it makes use of vector spaces does not make much sense. The vector space article should focused on its own concept, with a few illustrating examples, and trying to add to it a compilation of all concepts making use of it is close to insane. Even worse would be trying to do the same thing with the notion of topological space, or even just set (close to all of mathematics). PatrickR2 (talk) 21:48, 10 January 2023 (UTC)

I see what you mean and agree that some of that paragraph's information is better suited for the article "separated sets". I moved that information to that article. I left the sentence

Two subsets $S$ and $T$ are separated precisely when they are disjoint and each is disjoint from the other's derived set ${\textstyle S'\cap T=\varnothing =T'\cap S.}$

in this article.Mgkrupa 22:38, 10 January 2023 (UTC)

I argue that this sentence belongs in this article for the same reason that the sentence "A subset $S$ of a topological space is closed precisely when $S'\subseteq S,$" belongs in this article: they both add information about the notion of derived sets. Also, I'd like to point out that your argument against discussing the characterization of separated sets could also be used to argue against discussing this the characterization of closed sets:

"It is true that separated closed sets can be described in terms of derived sets (or equivalently in terms of closures). I am not objecting the validity of this fact in any way. My point is that it does not add any information to the notion of derived set, and if someone really cares about this fact (even though the equivalence between the two characterizations of separated closed sets is really a triviality), then it should be added to the article about separated closed sets and not here."

Finally, you dismissed mentioning this fact in part because it "is really a triviality" but I'd like to point out that this is not a valid argument since we editors should make technical articles and "Articles should be as accessible as possible to readers not already familiar with the subject matter" who are presumed to be new to the subject. It should be be written "one level down", which means:

• "consider the typical level where the topic is studied (for example, secondary, undergraduate, or postgraduate) and write the article for readers who are at the previous level."
• "articles on undergraduate topics can be aimed at a reader with a secondary school background, and articles on postgraduate topics can be aimed at readers with some undergraduate background."

Part of the reason why editors are instructed to "Avoid, as far as possible, useless phrases such as:

It is easily seen that ...

Clearly ...

Obviously ..."

is because something is obvious to you and I is not necessarily obvious to a person reading this article. The typical reader of this article is likely an undergraduate student for whom "the equivalence between the two characterizations of separated sets" might NOT be trivial. Mgkrupa 22:47, 10 January 2023 (UTC)

Thanks for moving some of the information to separated sets. I don't have access to Pervin. Did you check what Pervin actually says? Maybe he defines separated sets in terms of derived sets, maybe not. We should double check that. Also, maybe I am mistaken in that, but I would argue that Pervin is not a very well known source, not at the same level as classics like Dugundji or Willard or Engelking for sure. PatrickR2 (talk) 01:49, 11 January 2023 (UTC)

https://books.google.com/books?id=QpXiBQAAQBAJ&pg=PA52&dq=Pervin+foundations+of+general+topology+%22Hausdorff+Lennes%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjDqsy9tb78AhVyEVkFHXZvAqEQuwV6BAgFEAc#v=onepage&q=Pervin%20foundations%20of%20general%20topology%20%22Hausdorff%20Lennes%22&f=false He gives both formulations, at least in words. PatrickR2 (talk) 05:09, 11 January 2023 (UTC)

References