Talk:Meagre set

Meager sets and Borel hierarchy

The article claims that "meagre set need not be an F_σ set".  Maybe it would be good to provide a counterexample in the Examples section.  (I was not able to find a counterexample.)  --Kompik (talk) 14:23, 17 March 2009 (UTC)

Meh, it's not very enlightening, so I don't know that a counterexample would be worthwhile.  Here's one way to see that the claim is true:

• In a second countable space, there are (no more than) $2^{\aleph _{0}}$ open sets, because each open set can be identified with the sequence of basic open sets that forms it and thus identified with a sequence of natural numbers.
• Thus there are $2^{\aleph _{0}}$ closed sets, since each closed set is the complement of a unique open set and vice versa.
• Since every Fσ is determined by an ω-sequence of closed sets, this means that there are at most $2^{\aleph _{0}}$ Fσ sets.
• However, there is a meagre set of cardinality $2^{\aleph _{0}}$ (namely, the Cantor set) and every subset of a meagre set is meagre, which means that there are at least $2^{2^{\aleph _{0}}}$ meagre sets.  Thus at least one subset of the Cantor space is a meagre non-Fσ set.

It is also possible to construct non-Fσ meagre sets using tools of descriptive set theory related to the Borel hierarchy, but this requires more advanced techniques that would be out of the scope of this article.  — Carl (CBM · talk) 15:51, 17 March 2009 (UTC)

Dense and meagre

So rationals are both dense and meagre as a subset of reals? A clarification of this would illuminate a lot about meagre sets (IMO). —Preceding unsigned comment added by 128.122.80.95 (talk) 17:55, 13 August 2009 (UTC)

Yes, the rationals are both dense and meager in the reals. What is it about that that you feel needs to be clarified? Are you saying that we should simply state this fact somewhere in the article? It strikes me as not closely related to an exposition of meager sets in general, but if you think an example is needed to help people around some conceptual roadblock I'm not seeing, I wouldn't be opposed to that. But I'd like you to identify the roadblock so we know why we're doing it. --Trovatore (talk) 19:29, 13 August 2009 (UTC)

Not albsolutley necessary. Perhaps more important would be to show a proof or refer to a proof of the fact that rationals are meagre since this fact is mentioned in the article. —Preceding unsigned comment added by 128.122.80.95 (talk) 16:41, 14 August 2009 (UTC)

The proof is easy: since "a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X" and the rational numbers are countable it is sufficient to show that each individual rational number, as a subset of the reals, is nowhere dense. I.e. that its closure has an empty interior.

With the usual topology on the reals this is obvious. But note that it depends on the topology: with the discrete and trivial topologies it would be false since only the empty set would be nowhere dense. AlexFekken (talk) 10:10, 21 March 2012 (UTC)

Fat set

Is a non-meagre set in English also called a fat set? I know that it is (translatedly) called this in German, at least by my own lecturer on the subject... If this is English too, it should be noted somewhere, I think.--93.133.222.51 (talk) 19:06, 8 May 2013 (UTC)

Not in my experience. --Trovatore (talk) 19:33, 8 May 2013 (UTC)

Definition

I don't understand what is meant by "dense interior". Are not all interiors dense? Howard McCay (talk) 02:02, 21 June 2013 (UTC)

Suppose the underlying space is the reals. Then the interval [0,1], say, has interior (0,1), which is not dense in the reals. However, suppose S is the set obtained by starting with the whole real line and deleting the point 3. Then S is open (so it is its own interior), and also S is dense in the reals. Thus S has a dense interior.

Talk pages are supposed to be for improving the article, not answering general questions about the topic, but if this discussion exposes something that needs to be clarified in the text itself, then it's justified. What do you think would make the text clearer? --Trovatore (talk) 02:12, 21 June 2013 (UTC)

Banach Category Theorem

In any space X, the union of any family of open sets of the first category is of the first category.

This is either trivial, as there are no open sets of the first category, or deeply misleading. Yes, I understand that it is properly referenced; I suppose the author of that book uses nonstandard terminology. The Banach Category Theorem says that a set that is locally meagre is meagre. --Yecril (talk) 06:07, 11 October 2016 (UTC)

Incorrect Example (corrected)

The first of 'Meagre subsets and subspaces' under 'Examples and sufficient conditions' I believe has a mistake. Specifically the direction "only if" in "A singleton set is ... a non-meagre subset if and only if it is an isolated point" For a counterexample, consider the coarse topology with 2 points. The only 'nowhere dense' set is the empty set, thus the only meagre set is the empty set. However singletons are not open, therefore points are not isolated.

I am considering changing the line to "A singleton set is always a non-meagre subspace (i.e. it is a non-meagre topological space). If it is an isolated point then it is a non-meagre subset, while the converse holds for T₁ spaces.", with no reference (unless somebody points one out of course).

Ggshaw (talk) 20:52, 8 November 2020 (UTC)

I have now changed this.

Ggshaw (talk) 17:44, 20 November 2020 (UTC)

Undue weight given to meagre sets versus meagre subspaces

@Mgkrupa I recently streamlined the exposition and added a lot of information to this article and some related ones. One thing that I have not touched is the discussion about meagre sets vs. subspaces, but I moved it to its own subsection to make the rest more readable. It seems to me that too much weight is given to this distinction of sets/subspaces. None of the usual sources that I checked make a formal distinction of this; they rely on readers having enough familiarity with topology to make the distinction themselves. Even Narici does not. He only mentions subsets vs. subspaces in one sentence in one example, and not in any formal definition. (And by the way, the sentence A singleton is a nonmeager subset of a topological space iff the point is isolated quoted from Narici is not even a correct mathematical statement.) If you have a reference that makes this formal distinction, please let me know. Actually I think it's perfectly fine to mention this distinction, but not with so much emphasis. (If you want, I can write a draft of what I would have in mind and show you.) In addition, look at all the redirects that link to the article page: http://en.wiki.x.io/wiki/Special:WhatLinksHere?target=Meagre+set&namespace=&hidetrans=1&hidelinks=1. After seeing what you had done about this sets vs. subspaces, someone had the idea that we should have all these parallel links, some with sets, some with subspaces. This seems totally wrongheaded. Do you have any comments? PatrickR2 (talk) 04:36, 23 October 2022 (UTC)

Thank you for your improvements to the article. The section's point was to point-out and clarify an important technicality that is easily overlooked. Feel free to reword it in whatever way you feel will improve it. I didn't know about the new redirects and have no comment about them. Good luck. Mgkrupa 06:23, 23 October 2022 (UTC)

@Mgkrupa I came across this particular use of "meager subspace of E" in the context of TVS: https://books.google.com/books?id=Q-jvBwAAQBAJ&pg=PA117&dq=%22meager+subspace%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjR7oL67YP7AhW5FFkFHeiFAhYQuwV6BAgGEAc#v=onepage&q=%22meager%20subspace%22&f=false, exercise 12, and wanted to confirm with you. Do you think they use "meager subspace" there in the sense of linear subspace of E that happens to be a meager set in E, or in the sense of subset of E that is meager in itself when given the subspace topology? PatrickR2 (talk) 22:17, 28 October 2022 (UTC)

The statement is

Let $E,F$ be t.v.s. such that $E$ is a Baire space. If $H\subseteq {\mathcal {L}}(E,F)$ is not equicontinuous, show that $M=\{x:H(x){\text{ is bounded in }}F\}$ is a meager subspace of $E.$

For this statement, it depends on what "subspace" means: is it "topological subspace" or "vector subspace"? This ambiguity is unfortunate but in this case I think it means both, for this reason: $H$ not being equicontinuous will by itself guarantee that $M$ is a meager subset of $E$ (see Rudin Theorem 2.5 p.44 - $E$ does not need to be Baire to conclude this but all maps in $H$ must be continuous and linear, which is assumed). The set $M$ is always a vector subspace (regardless of whether or not $E$ is Baire or $H$ is equicontinuous; this is because scalar multiples, Minkowski sums, and subsets of bounded sets are again bounded and because $H$ consists of linear maps). So the only reason to include the assumption "$E$ is a Baire space" in the statement would be if it was needed to conclude that $M$ was a meager topological subspace of $E.$

There is a similar statement on p. 87

proof. Let $E_{M}$ be the linear hull of $M;$ $E_{M}$ is a non-meager subspace of $E,$ and hence a Baire space.

In this case (as on page 84) "non-meager subspace" means vector subspace + non-meager subset of $E$ + non-meager topological subspace (the third term follows from the second since every non-meager subset of every topological space is a non-meager subset of itself (p.9)). (BTW The Baire space conclusion is due to the fact that a TVS is a Baire space if and only if it is non-meager in itself (pp. 8 and 77)). Hope that helps. Mgkrupa 04:55, 29 October 2022 (UTC)

Thanks, that is quite helpful. But I would interpret the first statement quoted above (in Schaefer exercise 12 on p. 117) slightly differently. Rudin (Fun. Anal.) Theorem 2.5 says: if $M$ is nonmeagre in $E$, then $M=E$ and $H$ is equicontinuous. As an immediate consequence, (*)[if $H$ is not equicontinuous, $M$ is meagre in $E$], which corresponds exactly to the statement you quoted above (if one interprets "meager subspace of $E$" as meaning a vector subspace that is meager in $E$). As you mentioned, this does not use the hypothesis that $E$ is Baire. I don't know if the stronger result that $H$ would be meager in itself is even true in that case, but such a stronger result is not needed to show the rest of exercise 12 ("Thus if $\{H_{n}\}$ is a sequence of subsets of ${\mathcal {L}}(E,F)$ each of which is not simply bounded, ..."), provided one assumes that $E$ is a Baire space. Under that assumption the result follows easily from (*) only. So I think Schaefer put the assumption "$E$ is a Baire space" at the beginning of the exercise just for convenience, as it's needed to prove the rest of the exercise, and not to tell the reader to interpret the quoted sentence above as saying that "$M$ is a meager topological subspace of $E$".

As for the statement from p. 87, I agree with you interpretation. To summarize, I think that in the context of TVS Schaefer uses the terminology "meager/nonmeager subspace of $E$" to mean a vector subspace that is meager/nonmeager in $E$. That is respectively a weaker/stronger notion than a vector subspace that is meager/nonmeager in itself, that is, a vector subspace that is also meager/nonmeager as a topological space with the subspace topology. PatrickR2 (talk) 04:20, 30 October 2022 (UTC)

About Mgkrupa's latest edit (June 2023)

Hi @Mgkrupa. I saw your latest edits. First of all, thanks for preserving most of the changes (additional examples, properties, citations, etc) that I had previously made. I have a few smaller issues with yours though. I'll mention a few and you can comment. Then I'll mention more.

First of all, the reason I had moved the Examples section after the Properties section is that many of the examples can best be appreciated after one has read the properties, as they either depend on the properties or illustrate why some properties hold or don't hold is some of the hypotheses are changed. Why did you move the Examples back to before the properties/characterizations? PatrickR2 (talk) 19:00, 23 June 2023 (UTC)

Hello again @PatrickR2. I hope your doing well.

Answer 1: To answer your first question, let me quote Wikipedia:Manual of Style/Mathematics#Article body (boldface was added by me):

Representative examples and applications help to illustrate definitions and theorems and to provide context for why they might be interesting. Shorter examples may fit into the main exposition of the article, such as the discussion at Algebraic number theory § Failure of unique factorization, while others may deserve their own section, as in Chain rule § First example. Multiple related examples may also be given together, as in Adjunction formula § Applications to curves. Occasionally, it is appropriate to give a large number of computationally-flavored examples, as in Lambert W function § Applications. It may also be edifying to list non-examples, which almost-but-not-quite satisfy the definition. In keeping with the purpose and tone of an encyclopedia, examples should be informative rather than instructional (see WP:NOTTEXTBOOK for details).

I place the Example section immediately after the Definition section to help "illustrate definitions" since this is helpful for readers who have never encountered the article's definition before. After all, how useful would it be to list the properties of compact spaces (or nonmeagre sets, etc.) to a reader who is so new to the subject that they do not yet know whether or not $[0,1]$ or $(0,1]$ is/isn't compact and would have no idea what to say if they were to try to give a concrete example of a compact space? Not very. So if possible, a concrete example should be given as soon as possible. But of course, it is not necessary to put all concrete examples immediately after the definition (and there are even good reasons to not do this).

As an aside, I've been thinking of renaming the "Examples" section of this (and maybe other) articles to "Concrete examples" (right now, it consists almost entirely of concrete examples like $[0,1]$) or maybe "First examples". Less-concrete examples (e.g. sufficient conditions like "subsets of meagre sets are meagre") would be elsewhere. After all, properties, theorems, sufficient/necessary conditions are important and interesting. But concrete examples are essential to understanding the basics of whatever class of mathematical objects the article is discussing. Do you want to share your opinion on this?

Answer 2: You said:

many of the examples can best be appreciated after one has read the properties

A couple of points. (1) In general, such examples/counter-examples should (ideally) be placed as close as possible to the relevant property (no one likes having to jump around an article in search of something). And (2) if an example is a counter-example to something then this should be made clear to the reader unless (exceptions as always) there is some reason not to. Remember that readers do not memorize everything that they read (and often skip around the article) so it may not be clear to a reader that some particular example in the "Examples" sections is a counter-example to something mentioned once in some other far away section (e.g. "Properties").

Having the counter-example immediately after (or close to) the relevant statement (e.g. property, theorem) can also help YOU write less (and save you time) since it often allows you to avoid having to repeat parts (or even all) of the statement. For example, compare this:

Example close to statement: "[Some long unnamed technical result/proposition], although $[0,1]$ shows that the conclusion may be false if [this-or-that property] does not hold"

to the much wordier

Example far away from statement: "$[0,1]$ shows that if [repeat some technical details since they were mentioned somewhere far away] but [this-or-that property] does not hold then [repeat conclusion] may be false."

Finally, let's remember that it is NOT necessary to put ALL examples into the "Examples" section (and yes, I realize that this is exactly I did in this article - this is a less than ideal practice that I've made into a habit and I should stop - it seems that I should also start remembering this). In fact, if I remember correctly, it is recommended that examples be dispersed throughout the article. Mgkrupa 22:53, 23 June 2023 (UTC)

Thanks for the points that you made. After reading that, it actually makes sense and I agree with it. Like you said, it is good to have a section of simple examples immediately after the Definitions section, so that the reader can immediately solidify their understanding. And also have examples/counterexamples that illustrate particular properties together with the properties also makes it easier for the reader to grasp any subtleties. And, like you said, we don't need all of the examples in the Examples section. I would strongly recommend to keep the name "Examples" though, not change it to "Concrete examples" or "First examples". If other examples are scattered together with the properties, that's perfectly fine, even if they are not in the Examples section. So I guess what can be done is move some of the examples/counterexamples to be together with the corresponding properties that they illustrate and keep other ones in "Examples". Let me know if you want me to help with that, or you want to do it. PatrickR2 (talk) 06:08, 24 June 2023 (UTC)

I will start moving some examples around. PatrickR2 (talk) 22:31, 24 June 2023 (UTC)

Yes. Please feel free to start. (Sorry about the delay in my response but for some reason, Wikipedia did not notify me that you had replied). Mgkrupa 00:40, 28 June 2023 (UTC)

Or rather, I'll add more examples together with the properties. But first need to clear some things up. See issues below. PatrickR2 (talk) 14:09, 25 June 2023 (UTC)

Second general comment. I have noticed that many of the articles that you edit become (or at least it used to be the case) overly bloated sometimes (which seems to have bothered more than one person from past comments in WikiProject Mathematics). More specifically, say an article on topic A is being edited and there is a closely related concept B that also has an article of its own. You go in and start adding of lot of details about B in the article for A. That is not the right approach. The main thing is Wikipedia is not meant to "teach". It is meant to present facts in a systematic and organized way, but not meant to present things in a textbook like fashion where all the connections between A and B are presented together in the same place (assuming again that both A and B have their separate articles for whatever reason). Instead, article A can mention B and recall the definition of B or the main result for B, but without giving all these details about B, and can defer all the details for B via a link to the B article. That way, the article for A is focused on A and not diluted by extraneous considerations. If any reader wants or needs to know more details about B, it's just a click away. Again, the article about A is not meant to "teach" about all the other topics potentially related to A.

Now specifically for your changes, there is a separate article about B = nowhere dense sets. It's ok to recall that definition, as that's needed in the definition of meager set. But for example mentioning that it was called "rare" by Bourbaki is completely irrelevant in this article (that instead would belong to the B article, and it is already in that article). And in the same paragraph, you remind the definition of nowhere dense as being a set whose closure has empty interior. But then you immediately follow that with "closure(R) does not contain a non-empty open subset of X". That's ridiculous. Of course it's true, but there is absolutely no need for this bloat. You are just rephrasing the definition in a different way. But it's totally redundant. Again, this article is not meant to "teach" what nowhere dense set means. Or actually in this case, the only thing the rephrase does is explain what is meant by interior. But there is even less of a reason to "teach" this here in an article about meager sets. If a reader has no clue about what the interior of a set means, they have no business trying to understand meager sets. [That's actually another advantage I see about having less redundancies and explanation of basis concepts: it directs the unprepared readers to these building blocks via links, so they can better spend their time understanding the prerequisite concepts over there, which allows the more advanced article to be more focused, less bloated and easier to read (for readers with the right prerequisites).] One could also analyze each of the sentences in the rest of the paragraph, but it's all best left to the B article where they belong. Looking forward to hear your thoughts. If you don't disagree, I'll simplify that whole paragraph. PatrickR2 (talk) 02:36, 25 June 2023 (UTC)

Here is another small example of the same symptom. You replaced what I had before: The empty set is a meagre subset of every topological space. with The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space. The word "always" is completely redundant here. It adds no value. Wikipedia is supposed to present facts in an encyclopedic way and not to "teach" people who don't know the subject. Also the article is focused on the meager property. So the fact that the empty set is closed is irrelevant and should be left out. It is even misleading, as it seems to imply that the set is meager because it is closed and nowhere dense. It would be equally meager if it were nowhere dense and not closed. I could find many other similar examples, which is one thing I had take care of cleaning in my previous edits. I don't understand why you want to put all that back. PatrickR2 (talk) 14:04, 25 June 2023 (UTC)

The word "always"

This word is redundant but I included it since according to MOS:MATH#TONE "Mathematics articles are often written in a conversational style similar to a whiteboard lecture." This word can be removed. But in general, I've come to learn that it's best to explain

So the fact that the empty set is closed is irrelevant and should be left out.

Since you pointed it out, I now realize that I should have worded that better. Since this was the first example of a meagre set given in the article, I wanted to give a brief explanation (in a conversational style) of why it is a meagre set (remember that we should write one level down WP:ONEDOWN and that what is obvious to you and me might not be obvious to someone just learning Topology). Mgkrupa 01:03, 28 June 2023 (UTC)

As you were not responding for a while, I have gone ahead and removed the bloat related to nowhere dense sets. That makes the article more streamlined and focused, easier to read and digest. But feel free to discuss further here. Also, about the example of the empty set above. Another illustration of bloat that we should avoid. PatrickR2 (talk) 21:32, 26 June 2023 (UTC)

As I mentioned in a reply above (a few minutes ago), Wikipedia did not notify me that you had replied. My apologies for the delay. Mgkrupa 01:05, 28 June 2023 (UTC)

I do unfortunately have a tendency of making additions that are more similar to textbooks (WP:NOTTEXTBOOK) than encyclopedias. This is something I'm working on and so I appreciate it when this is pointed out to me (as you did above).

"If any reader wants or needs to know more details about B, it's just a click away."

I give additional details like that since per MOS:LINK: "Do not unnecessarily make a reader chase links: if a highly technical term can be simply explained with very few words, do so."

"But for example mentioning that it was called "rare" by Bourbaki"

So for that particular addition, I wanted to include the origin of this term since I had just read it (in the source) and so it was easy to add a citation + page number. I knew at the time I added it that it should instead be added to the article Nowhere dense set but did not have the energy at the time to make those changes (I'd have to read through the article to find the right place to add this fact, padd the reference to bibliography, etc.). I intended to move this fact to the article Nowhere dense set later but forgot. Oops. Mgkrupa 01:25, 28 June 2023 (UTC)

There is no need to add "rare" to the Nowhere dense set article, as it has been in there since 2021. Also, the word "rare" was used in French in the French original edition of Bourbaki. But the version of Bourbaki in English never used "rare", it used "nowhere dense". As this is an encyclopedia in English, there should be no undue emphasis on the word rare. (Compare that to the definition of a concept that first appeared in German or Russian or whatever at the beginning of the 20th century for example, and we don't use the German/etc word for it, but the corresponding English word.) The fact that Bourbaki used "rare" in French is irrelevant. Now the reason we need to mention "rare" is that some people in English (influenced by Bourbaki) started using that word for that same concept, particularly among functional analysts as I understand. That is a valid reason to include it. And probably the reason they did so is that a single adjective is grammatically simpler to use that a compound expression like "nowhere dense". PatrickR2 (talk) 03:03, 28 June 2023 (UTC)

Above you also mention MOS:MATH#TONE: Mathematics articles are often written in a conversational style similar to a whiteboard lecture. But you are taking this out of context. the full quote is: Mathematics articles are often written in a conversational style similar to a whiteboard lecture. However, a narrative pedagogical style runs counter to Wikipedia's recommended encyclopedic tone. In other words, we are not supposed to use conversational style here. PatrickR2 (talk) 03:12, 28 June 2023 (UTC)

Are you sure that "conversational style" and "narrative pedagogical style" are synonyms? Mgkrupa 16:07, 28 June 2023 (UTC)

@Mgkrupa With the two sentences together, I read them like this. The first sentence says that mathematics articles are often written with a certain style (without saying that it should be done that way or not). It also seems to refer to mathematics articles in general, for example expository articles in journals like Americal Mathematical Monthly, etc, not just Wikipedia articles. If after reading the first sentence, one could interpret as saying we have to follow that style, the second sentence immediately dispels that notion by saying don't do that. (Notice how to the two sentences are linked by the word "however" to make that clear.) In other words, whether "conversational style" and "narrative pedagogical style" are the same or not, we are told: don't use conversational style, don't use whiteboard lecture style, don't use narrative pedagogical style. PatrickR2 (talk) 19:11, 28 June 2023 (UTC)

Also about MOS:LINK: Do not unnecessarily make a reader chase links: if a highly technical term can be simply explained with very few words, do so. This is exactly what I have done. I used to just have the link to nowhere dense, but I compromised and added the definition, as a reminder to the reader. If a reader wants more than that, it's a click away.

As I see it, the purpose of Wikipedia is not to predigest all the information to teach someone who does not know anything. It's to present the information in a coherent fashion with references, and also without an overwhelming amount of minute details. The purpose is to provide the reader with an overview of a topic and then the reader can follow up on the references if they want a deeper understanding. This can of course be debated, but I am pretty sure it's somewhere in one of the MOS or other guidelines.

(As a side note, even though the purpose of Wikipedia is not to "teach", of course people can learn a tremendous amount of things from it and that's all to the good. But regarding learning something, a reader can sometimes learn more if less is said, so that the reader has to think by himself about something, instead of just reading the explanation. About the empty set example again, just saying "the empty set is meager" will make the reader wonder why, they will go back to the definition and see the notion of "nowhere dense", then ask themselves "is the empty set nowhere dense", then ask themselves about the closure of the empty set, etc, etc. That way, things will stick for them much better than just reading that it's nowhere dense, hence meager. That's the reason I had removed the details from a previous version that you had in 2022/2021?. And that's what I meant by "less is more". But no big deal. If we want to say the empty is nowhere dense, that's fine too.) PatrickR2 (talk) 03:36, 28 June 2023 (UTC)

As I see it, the purpose of Wikipedia is not to predigest all the information to teach someone who does not know anything. It's to present the information in a coherent fashion with references, and also without an overwhelming amount of minute details. The purpose is to provide the reader with an overview of a topic and then the reader can follow up on the references if they want a deeper understanding.

Well said. I'll try to remember that advice. Mgkrupa 16:09, 28 June 2023 (UTC)

@Mgkrupa mentioning your name, so that you are notified. Not sure if it's needed, but maybe that's why you were not notified before? PatrickR2 (talk) 03:40, 28 June 2023 (UTC)

This worked. Wikipedia notified me of this reply but not any of your other replies. Mgkrupa 16:10, 28 June 2023 (UTC)

Third general comment. In my previous edits from 2022 I had carefully removed a lot of duplication. Now with your latest edits many of these duplications have been reintroduced. As a heads up, unless you have a valid objection, I plan to re-eliminate these duplications (and if there is debate about any of the changes, I will give detailed justification of each change if you think it necessary, explaining why it was redundant.) PatrickR2 (talk) 20:06, 28 June 2023 (UTC)

@Mgkrupa Also, your changes reintroduced an incorrect statement of the "Baire category theorem". PatrickR2 (talk) 20:09, 28 June 2023 (UTC)

"I had carefully removed a lot of duplication. Now with your latest edits many of these duplications have been reintroduced."

Then next time, don't remove inline citations and don't mix unsourced statements into sourced sentences, which was the reason I restored that content. Mgkrupa 21:18, 28 June 2023 (UTC)

"I plan to re-eliminate these duplications"

Go ahead, but remove the duplicates that do not have citations. Mgkrupa 21:26, 28 June 2023 (UTC)