Talk:Dense-in-itself

Article has a serious problem

This article is an example of what can go wrong when a Wikipedia article is simply copied from another source (in this case, PlanetMath). The result in this case is the the Wikipedia article is somewhere between impossible to understand, and wrong (it's hard to tell where).

Here is the beginning of the article:

"In mathematics, a subset A of a topological space is said to be dense-in-itself if A contains no isolated points.

"Every dense-in-itself closed set is perfect. Conversely, every perfect set is dense-in-itself.

"A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers. This set is dense-in-itself because every neighborhood of an irrational number x contains at least one other irrational number . On the other hand, this set of irrationals is not closed because every rational number lies in its closure. For similar reasons, the set of rational numbers is also dense-in-itself but not closed."

Never mentioned, but apparently playing a big role here, is the set of real numbers. It is only in relation to the set of real numbers that it is obvious the last sentence, about the set of irrational numbers, is true.

So the article should make crystal clear that "Given a topological space X and a subspace A, then [etc., etc.]." And the last sentence should say something like "The irrational numbers, considered as a subspace of the reals, [etc., etc.]."

For, there are many topological spaces X for which the space J of irrational numbers (with the subspace topology from the reals) forms a closed subspace of X. (E.g., consider X = J itself.)Daqu (talk) 20:11, 30 July 2010 (UTC)

I'm not sure if that's really a *serious* problem. It's an easy fix to make the language a little more rigorous, and I think the intent is clear. In any sense, it's easy to fix.Rodya mirov (talk) 20:26, 17 October 2012 (UTC)

Bullet lists should be avoided

PatrickR2 reverted this edit of mine that changed a bullet list into paragraphs saying:

Undid revision 1132822549 by Mgkrupa (talk) reverting vandalism of the Properties section. Your reorganization is less clear.

I think that calling it vandalism is a little over the top. But anyways, bullet lists are not encyclopedic and goes against Wikipedia's manual of style. See MOS:LISTBULLET and Wikipedia:Manual of Style/Lists. In particular, MOS:LISTBULLET states:

Do not use lists if a passage is read easily as plain paragraphs.

@PatrickR2: since you don't approve of my edit, please change the article so that it complies with MOS:LISTBULLET and Wikipedia:Manual of Style/Lists. Mgkrupa 00:34, 12 January 2023 (UTC)

@Mgkrupa: I see that David Eppstein has since made the change to remove the bullet list. From https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics it seems that other people are also of the opinion that lists are not the best way to present information. You mentioned that you used to make lists and changed your approach later on, which resulted in better contributions. So you guys may be right about that, and I am fine with that and will try not to overuse lists in the future.

It also was a little harsh to call your change "vandalism", but the other reason I did so is that we had two sections, one about Examples and one about Properties. You unilaterally moved most of the properties to a new examples section combined with "necessary conditions". In my opinion that is not a better way to present information. At least now we still have Examples and Properties, which is quite clear. PatrickR2 (talk) 23:40, 12 January 2023 (UTC)

"I am fine with that and will try not to overuse lists in the future"

I'm glad to see that your open to constructive criticism and trying new things. I hope that your experience with avoiding lists is as beneficial to you as it was for me. Also, I have thick skin so the vandalism label didn't affect me much. If ever in the future there is something that you want to ask or tell me then please always feel free to do so.

"You unilaterally moved most of the properties to a new examples section combined with "necessary conditions". In my opinion that is not a better way to present information."

So let me explain some of my reasoning for making those changes. Assuming that this article is to have an "Examples" section and a "Properties" section, then how should we decide whether a given sentence belongs to "Examples" or to "Properties"? Since this subject of this article is "dense-in-itself sets", a "property" of a dense-in-itself set should be (by definition) an attribute or quality of a dense-in-itself set (EX: IMO sentences of the form

If A is a dense-in-itself set [...] then ${\underline {~~\;~~}}$ is true of A

belong to the properties section); whereas a sentences belongs in the "Examples" section if it can naturally/easily be made to fit the form:

[...] Then ${\underline {~~\;~~}}$ is a dense-in-itself set.

For example, using this rule, the following (taken from here) belong in the "Examples" section:

• "The union of any family of dense-in-itself subsets of a space X is dense-in-itself."
• "Every open subset of a dense-in-itself space is dense-in-itself."
• "Every dense subset of a dense-in-itself T1 space is dense-in-itself."

However, the following doesn't belong to the "Examples" section

• "In a topological space, the closure of a dense-it-itself set is a perfect set."

because it can not easily be made to fit the form "[...] then ${\underline {~~\;~~}}$ is a dense-in-itself set". Instead it belongs to the properties section since it can be reworded to fit the form: "If A is a dense-in-itself set then ${\underline {~~\;~~}}$ is true of A" (specifically, it can be reworded: "If A is a dense-in-itself set then its closure is perfect")

Did I explain this in a way that makes sense? Mgkrupa 23:59, 15 January 2023 (UTC)

Now I see where you are coming from. Let's say we have a wikipedia article about definition ABC (a property of topological spaces as an example). In my mind (I always thought that was the usage of such sections in wikipedia articles), the Properties section would contain what would be labeled in textbooks as propositions and theorems. That includes both things like "any space satisfying XYZ (together with other things maybe) also satisfies ABC" (with ABC in the conclusion), and "any space satisfying ABC (and maybe other things) satisfies XYZ" (with ABC in the hypotheses). It can even be of a more general form, basically any result having ABC in its statement ("the homotopy type of the product of an ABC space and ... is not ..."). In contrast, an Examples section presents concrete examples and counterexamples. For topological properties, that would be a specific topological space or subset, showing that it does or does not satisfy ABC, or illustrating a specific theorem involving ABC, or showing if some of the hypotheses are relaxed, a result is not true anymore, etc, etc.

(Side note: People may have different opinions about this, but I think that there are cases where it is clearer to have Examples after the Properties. If the reader has seen a mention of all the properties first, it's easier to illustrate the theorems when presenting examples (and sometimes the same example can illustrate multiple theorems). It would be hard to illustrate the theorems if they have not been mentioned yet. (Alternatively, add illustrating examples together with some theorems, but sometimes that clutters the presentation and it may be clearer to have the examples separate and mention to the reader that details are further in the examples section.) But I also think that very often it's perfectly fine to have Examples first. It all depends on the article.) PatrickR2 (talk) 06:51, 16 January 2023 (UTC)

Personally, I like it when examples are dispersed throughout the article (and if I remember correctly, this is also recommended by the Math Manual of Style). For example, placing a relevant example/counter-example next to a fact/property is IMO more helpful than placing it in a distance "Examples" section, although doing this "cleanly" (i.e. so that the resulting paragraph doesn't read awkwardly) is frequently easier said than done (and consequently, often ends up not being done). All other examples goes in the "Examples" section (hopefully they won't stay there though) although I've noticed that even for such an example, it's often possible to add a sentence or two explaining why it's important, which in turn sometimes makes it much easier for me to find it a home outside of the "Examples" section. Mgkrupa 16:41, 16 January 2023 (UTC)

Regarding a Properties section, based on what I have seen on many articles I was using it with an informal meaning, i.e., any kind of result involving the property ABC. Any mathematically true statement could be viewed as a "property" in that sense: the statement $\forall x(\phi (x)\implies \psi (x))$ where $\phi$ and $\psi$ are some formulas with free variable $x$ in some theory can be viewed as saying "all objects (in an appropriate universe of discourse) satisfying $\phi (x)$ have the property that they also satisfy $\psi (x)$". Do you think Properties sections are usually used in a much more restrictive sense? PatrickR2 (talk) 01:23, 17 January 2023 (UTC)

I don't know of any rules or conventions about what does or doesn't belong in the "Properties" section (maybe MOS:MATH or WP:Make technical articles understandable mentions something?). It seems to vary a lot from one article to the next. Mgkrupa 02:40, 17 January 2023 (UTC)

@David Eppstein: Thanks for changing the Properties section from a bullet list to paragraphs. Looks good, except that you introduced a mathematical mistake (regarding what used to be the second bullet). Would you mind fixing it? PatrickR2 (talk) 23:43, 12 January 2023 (UTC)