Talk:Perfect set

question
Is this correct? An isolated point forms itself an open set. How does that affect whether every point is a countable intersection of open sets?--Patrick 15:31, 13 June 2007 (UTC) I'm sure it's wrong. An open set is certainly a g-delta and so a discrete space has all points g-delta even though it is as far from perfect. I'll remove itA Geek Tragedy (talk) 16:27, 30 May 2008 (UTC)

suggestions for improvement
Most of what I am going to say here is not in the true/false dichotomy, but is a matter of perspective, and therefore open to some degree of interpretation. The big picture is that I think the article misses the main point. The article lists many properties and non-properties about perfect spaces. But what is the main point about perfect spaces? As far as I understand (and I do not claim my understanding to be complete) the main point is that there exist totally disconnected perfect spaces. Like other results in point set topology, this has to do with exploring how bizzare things can get. (The standard example of a perfect space that is totally disconnected is not so very exotic, just our familiar Cantor set.)

In fact, the set of all rationals with the standard subspace topology (from the reals) and the set of all real numbers with the lower limit topology are both perfect and are totally disconnected. Also, there is nothing bizzare about the fact that there are totally disconnected spaces that are perfect (in my opinion). This is why I emphasized that perfect spaces aren't really 'perfect' so technically I did explore what you suggested.

Topology Expert (talk) 04:29, 6 July 2008 (UTC)

As far as I am aware, the notion of perfect space is not "useful" like the concept of compactness. By this I mean that I don't think it ever happens (or perhaps very seldomly) that you would prove that a space is perfect as a step in order to prove something else.

No. You wouldn't do this but sometimes proving a space is perfect can be used as a 'preliminary'. For example, before verifying that a space is simply connected, you would first verify that it is path connected before checking its fundamental group.

Topology Expert (talk) 04:29, 6 July 2008 (UTC)

The lede is very problematic. Usually, the lead should explain the main point of an article informally without going into technicalities and details. But in this case the lede is misleading. For me "generally large enough" is not any imprecise approximation to some fact about perfect spaces. The same applies to "more likely to be connected". Perhaps the sentence "Note however, perfect spaces are generally not substitutes for spaces satisfying other important topological properties" is meant to say something similar to what I have written in the last sentences of the previous paragraph, but that's really not clear to me.

A non-perfect space is more likely to be disconnected (therefore a perfect space is more likely to be connected). This is because a non-perfect Hausdorff space with more than one point is disconnected as indicated in property 6 (note that just adding the Hausdorff condition makes it disconnected along with the condition that the space must contain more than one point). The use of 'more likely' is quite vague but still gives the reader an impression that perfect spaces are useful in some respect (which is true). Also, the name 'perfect' can be misleading and can compel people to believe that perfect spaces satisfy 'many' useful topological properties. By writing what I did in the introduction, I am showing that this is in general not the case.

Topology Expert (talk) 04:29, 6 July 2008 (UTC)

In item 6, the assumption that X contains more than one point is missing.

I admit I made a mistake here. Therefore, I fixed this.

Topology Expert (talk) 04:29, 6 July 2008 (UTC)

Instead of Theorem 1, it is reasonable just to say "By the Baire category theorem a perfect compact Hausdorff space is uncountable."

Actually, theorem 1 doesn't follow from the Baire category theorem but the proof is similar to the Baire category theorem. Note that both theorems use the finite intersection property but this doesn't imply that they are dependent on each other.

Topology Expert (talk) 04:29, 6 July 2008 (UTC)

Theorem 2 is not really a theorem. It is an obvious fact.

Yes. It is an obvious fact and perhaps it would be better to include it in the 'examples and properties' section.

Topology Expert (talk) 04:29, 6 July 2008 (UTC)

Oded (talk) 05:45, 5 July 2008 (UTC)

Perhaps more important than all of my comments above: can you provide citations indicating that all these facts are known and meet the WP:V and WP:RS standards? Oded (talk) 18:01, 5 July 2008 (UTC)

Dear Oded,

I am supplying the proof of each and every property (the proofs in general lack minor details but should serve as verifications of each property)

Proof of 1:

The proof is already supplied in the article.

Proof of 2:

Since every subspace of X is perfect, so are two point sets (since X has more than one point, there are two point subsets of X). Therefore, since a two point set is not perfect, it must have the indiscrete topology so that one point sets cannot be closed.

Proof of 3:

Suppose a one point set is open relative to the product of arbitrarily many perfect spaces. Since the projection maps are open (with respect to both the box and product topologies), each co-ordinate of this point must be open relative to the respective space; a contradiction since each component space is perfect.

Proof of 4:

The proof is already supplied in the article.

Proof of 5:

The proof is already supplied in the article.

Proof of 6:

If X is a non-perfect Hausdorff space, then there is at least one isolated point in X. Since one-point sets are closed in X, this isolated point is also closed. Since there is a subset of X which is both open and closed (the isolated point), and since X has more than one point, X is disconnected. The proof is exactly the same if X is a non-perfect, T_1 space with more than one point.

Proof of 7:

An infinite set in the indiscrete topology is not Hausdorff and therefore cannot be metrizable; a space in the discrete topology is metrizable (the discrete metric induces the topology of this space) and is clearly not perfect.

Proof of 8:

A finite subspace of a T_1 space always inherits the discrete topology and is therefore not perfect.

Proof of 9:

If X is a topological space and if {x} is an isolated point of X, then X is locally connected at x. Similarly, X is locally compact at x. The rationals is perfect but not Baire; the set of all real numbers in the lower limit topology is perfect but not second countable; R^(w) [the countable product of R with itself] given the box topology is perfect but not first countable.

Proof of 10:

The only metric topology on a finite point set is the discrete topology and the discrete topology is not perfect.

Proof of 11:

If G is a topological group having at least one isolated point (call it x), then consider the map f(z) = a*z for some element 'a' in G. It is a standard fact that 'f' is a homeomorphism (see Munkres, chapter 2 in the section entitled supplementary exercises on topological groups). If 'y' is an element of G then let a = y*(x^(-1)). Then f(x)=y and since f is a homeomorphism, {y} must be open.

Proof of 12:

If (G_a) is an open cover of (X, T) [where T is the indiscrete topology on X), then the collection (G_a) must contain X (the whole space) so that X itself is an open refinement of order 1. Therefore, the topological dimension of X is 0.

I have already given proofs of the theorems. I am quite sure that these proofs are correct but if you do come across any mistakes please let me know. Please also ignore the fact that in some proofs, some details may be left out.

Topology Expert (talk) 05:11, 6 July 2008 (UTC)

Dear Oded,

Please read what I have written in response to your comments. I have also given proof of every single fact. This should serve as a verification.

Thanks

Topology Expert (talk) 04:29, 6 July 2008 (UTC)


 * Dear Topo, You are right, of course, in your comment regarding the rationals being a totally disconnected perfect space. The point about the Cantor set is that it is also compact and Hausdorff (so perhaps slightly more surprising).


 * The point I was making with regards to sources was not in order to verify correctness. WP has the WP:OR policy. This means in particular that material you include has to be accompanied with references. If you would cover some material and add some reasonably short illustrative example without a reference, I would probably not complain. But here the bulk of the article consists of a list of 12+ facts, which seem suspect of being original research.


 * With regards to the Lede - I am unconvinced. But since we had too many disagreements recently, I will not insist on this right now. Instead, I'll try to focus on what it is likely that we can agree on.


 * I will reply with regards to "Theorem 1" shortly. Oded (talk) 16:50, 6 July 2008 (UTC)


 * Incidentally, I think iff is considered too informal to be used in WP articles (but I'm not sure). You might want to check. Oded (talk) 17:01, 6 July 2008 (UTC)

Baire catogory theorem implies "Theorem 1"
For every $$x\in X$$ let $$V_x:=X\setminus \{x\}$$. Since $$X$$ is perfect, $$V_x$$ is dense. Since $$X$$ is Hausdorff, $$V_x$$ is open. If $$X$$ would be countable, then the Baire category theorem would imply that $$W:=\bigcap_{x\in X} V_x$$ is dense in $$X$$. But $$W=\emptyset$$, which is not dense in $$X$$. (I guess it is taken for granted that $$X=\emptyset$$ is not allowed. Perhaps this should be made explicit.) QED Oded (talk) 17:01, 6 July 2008 (UTC)

Dear Oded,

Your proof is correct and I guess I didn't believe you earlier. Perhaps you should include this proof in the article. Before creating an account, I used to read Wikipedia for definitions in mathematics and quite often I did come across the use of iff so this is why I included it in the article. I also didn't write that perfect spaces are 'generally large enough'; I think Jack wrote that.

Thanks

Topology Expert (talk) 01:46, 7 July 2008 (UTC)


 * I think before further progress on the article can be made, the WP:OR issue has to be resolved. Oded (talk) 06:57, 8 July 2008 (UTC)

Also, note that if a space has 'more' (which cannot be made mathematically precise in the context of infinite sets unless you talk about countability and uncountability) open sets, it is more likely to be Hausdorff and less likely to compact. It is also less likely to be perfect. Basically this gives a rough idea to the reader of what types of spaces are perfect (the ones with fewer open sets although this doesn't guarentee that they are perfect). I guess that perhaps it is not necessary to include this in the article but this can be easily removed.

Topology Expert (talk) 01:50, 7 July 2008 (UTC)

I got to go now, but have a look. Never mind, though, it really does not matter who wrote this. More later. Oded (talk) 01:55, 7 July 2008 (UTC)

Terminology issue
I am concerned that the definition of perfect space used in this article is not the correct (i.e., standard) one. Elsewhere on wikipedia are the following two terms:


 * A perfect set is a subset of a topological space which is equal to its own derived subset, i.e., it is a closed subset without isolated points. Note that this is a property of subsets rather than spaces (although in the context of e.g. metrizable spaces it makes sense to call a compact space without isolated points a perfect set since it will automatically be closed in any Hausdorff space that contains it).


 * A perfectly normal space is a space for which any two closed subsets can be precisely separated by a continuous function. Equivalently, it is a normal space in which each closed subset is a G-delta set.

All references for "perfect space" I have been able to find in the literature (e.g. Engelking, General Topology, p. 48) mean a space for which every closed subset is a G-delta set. (Thus perfectly normal = perfect + normal. Perhaps this was the origin of the term.)  Since any metrizable space is perfectly normal, this is obviously an inequivalent use of the term. Moreover, I have certainly heard "Let X be a topological space without isolated points" many, many times: i.e., it is not deemed necessary to give this concept a special name.

Therefore I submit that the present article is devoted to a sequence of facts about spaces without isolated points, whereas it should rather be discussing the (somewhat less immediately transparent) concept of spaces in which every closed subset is a G-delta set.

If someone disagrees, please give references for this use of perfect space: since I can find many references for the other use of perfect space, more than one or two would be preferred. Plclark (talk) 16:04, 11 July 2008 (UTC)Plclark


 * Assuming that the term perfect space does not appear in the literature (or hardly does), I think that the present article should be deleted, and perhaps a (short) article perfect set might be created. Perhaps simpler would be to move the present article to perfect set, and then rewrite it including only quotable and significant facts. Oded (talk) 19:56, 11 July 2008 (UTC)

Searching for perfect space on MathSciNet yields 97 hits. A cursory inspection seems to indicate that at least half of these refer to the "closed sets are G-delta sets" definition. I did not find any which clearly refer to the "no isolated points" definition although of course one would have to look harder to be sure. So as far as I can see the above assumption may not be warranted. I am not a general topologist, though; it might be nice to have one weigh in on the matter. Plclark (talk) 21:15, 11 July 2008 (UTC)Plclark

Dear all,

According to Rudin's 'Principles of mathematical analysis' (third edition), page 32, definition 2.18 (h), a subset E of a metric space X is perfect if E is closed and if every point of E is a limit point of E (i.e E has no isolated points). Note that in a metric space, every closed set is a Gδ set.


 * I did note this in my above comment. Plclark (talk) 10:55, 18 July 2008 (UTC)Plclark

In particular this implies that the definition you found is not equivalent to Rudin's definition for metric spaces (Rudin requires the additional condition that the metric space has no isolated points; according to your definition every metric space is perfect). However, the definition in Rudin's does imply yours, i.e Rudin's definition is stronger that yours for metric spaces.


 * And this as well. Plclark (talk) 10:55, 18 July 2008 (UTC)Plclark

I just had a glimpse at chapter 2 of Rudin's book and I noticed that many of his proofs about compactness in metric spaces inexplicitly use the fact that metric spaces are Hausdorff (for instance theroem 2.34, page 37). In fact most of his theorems about compactness do generalize to arbitrary Hausdorff topological spaces. Therefore, I am led to believe that despite the fact that Rudin only considers metric spaces in the definition of perfectness, his definition probably does apply to arbitrary topological spaces.


 * I don't follow you here. This is an instance of the perfect set definition cited above.  Rudin does not define even a metric space itself to be perfect, only a subset of a given space.  Again, this is mentioned above.  Plclark (talk) 10:55, 18 July 2008 (UTC)Plclark

Note also that I was not the one to give the definition of a perfect space; I expanded upon the definition.


 * True. But your expansion was not based on any referenced material, which is an OR issue. In this case the likely effect will just be that this material will be removed (or possibly moved to perfect set) and replaced with the sourceable (and therefore "correct") definition of perfect spaces as above.  Plclark (talk) 10:55, 18 July 2008 (UTC)Plclark

The creator of the article defined a perfect space as a topological space that has no isolated points. I think that it is best to question the creator as to why he defined perfect spaces in this way. Most probably, he referred to Rudin's book when giving the definition. Topology Expert (talk) 11:11, 17 July 2008 (UTC)


 * It might be appropriate to contact the creator, but it is not clear that he/she is still active on wikipedia. Many people make their edits and then move on (not the most desirable state of affairs, necessarily, but a wikipedic reality).  In any case, of course the talk page is the best place to discuss changes to the article in question.  Plclark (talk) 10:55, 18 July 2008 (UTC)Plclark

The issue is most likely just that "perfect" is used to mean many things. Two books that use the same terminology as this article are Classical Descriptive Set Theory, Kechris, and Basic Set Theory, Levy. You can search for "perfect Polish space" on google books to find many more examples. Since a Polish space always has the property that a closed set is G&delta;, the adjective perfect there always refers to the lack of isolated points.

I'll have to wait until tomorrow to look for references on spaces where every closed set is G&delta;. I am pretty confident that Counterexamples in Topology has some relevant examples, it may have a term for them. &mdash; Carl (CBM · talk) 22:59, 3 August 2008 (UTC)


 * I looked on MathSciNet and found 18 references for "perfect Polish space". This suggests that when one is discussing Polish spaces only it would be reasonable to use perfect in this way, especially since Polish spaces are (by definition) metrizable and any metrizable space is one in which closed subsets are G-delta sets.  However, this article presents this definition on the class of topological spaces, which seems not to be how the terminology is used.  Here is another reference supporting this definition (or, rather, an obviously equivalent one):


 * Pareek, C. M.(3-AB)
 * Perfect spaces.
 * Questions Answers Gen. Topology 9 (1991), no. 2, 151--157.
 * 54D20


 * A space $X$ is said to be perfect if every open set in $X$ is an $F_\sigma$-set. It is shown that a space $X$ is perfect if and only if for each countable collection $\scr U=\{U_i\colon\ i\in\bold N\}$ of open subsets of $X$ there exists a sequence $\{\scr G_j\colon j\in\bold N\}$ of open covers of $X$ such that $x\in U_i$ for $i\in\bold N$ implies there is $j\in\bold N$ such that $\roman{st} (x,\scr G_j)\subset U_i$. As a consequence of several other characterizations obtained by the author it is shown that: (1) a space $(X,\tau)$ is perfect if and only if for every countable collection $\scr C$ of open subsets of $X$ there is a topology $\tau_{\scr C}$ such that $\scr C\subset\tau_{\scr C} \subset\tau$ and $(X,\tau_{\scr C})$ is a developable and Lindelöf space; (2) a space $(X,\tau)$ is perfect if and only if for every $\sigma$-point finite collection $\scr D$ of open subsets of $X$ there exists a topology $\tau_{\scr D}$ such that $\scr D\subset \tau_{\scr D}\subset\tau$ and $(X,\tau_{\scr D})$ is a metacompact developable space (the author asks whether one could prove these two results for point countable and $\sigma$-locally countable open collections). It is also shown that if $X$ is $\sigma$-metacompact and perfect, then $X$ is $d$-paracompact.


 * Plclark (talk) 23:56, 3 August 2008 (UTC)


 * It seems to me that (as is too common) the same terminology is used different ways. I think we agree there are two concepts:
 * A space with no isolated points
 * A space in which every closed set is G&delta;, or equivalently in which every open set is F&sigma;
 * Both are well established properties that we should have articles on. Each article will have to list the terms used for its concept. The main question is what to actually name the articles. So what names would you propose that we use for the articles? &mdash; Carl (CBM · talk) 00:43, 4 August 2008 (UTC)


 * P.S. I noticed on google books that Open Problems in Topology II, Elliot Pearl 2007, p. 701 defines a "perfect space" to be one with no isolated points. I thought for a moment that definition might be limited to set theorists, but apparently not. &mdash; Carl (CBM · talk) 00:52, 4 August 2008 (UTC)


 * Unfortunately I cannot get google books to show me that page, and my university library was closed today. I did notice that in the first article in the same book the term perfect space is used without definition, and after enough reading it became clear to me that they mean the G-delta set definition.  So it looks as if working topologists are willing to tolerate multiple meanings.


 * At the moment it seems to me that the appropriate thing to do is to have two different pages on perfect spaces, one of them titled something like "perfect space (no isolated points)" and the other "perfect space (closed subsets are G-delta sets)". This article can be renamed to be the former, and G-delta space can be renamed to be the latter (especially since I have found zero citations for the use of the term "G-delta space").  How does this sound?[User:Plclark|Plclark]] (talk) 10:11, 8 August 2008 (UTC)


 * I also haven't been able to find any references for the term "G&delta; space". The references I did find only worry about that property in the context of perfectly normal spaces. The names you suggest are fine with me. Then the page Perfect space would become a disambiguation page that links to the two meanings. &mdash; Carl (CBM · talk) 11:42, 8 August 2008 (UTC)


 * Richard Pinch give me a reference for G-delta spaces that I had mistakenly overlooked - Counterexamples in Topology, p. 162. I added that source to the other article. I have added a clear explanation to the lede of each article, which should help a reader who is looking for the other meaning. Since we do have references for all the terminology now, and the current titles are much more likely to be searched for than the longer ones, I no longer am in favor of renaming the articles. &mdash; Carl (CBM · talk) 19:42, 8 August 2008 (UTC)

Let me return to the first point of this section:
 * A "perfect set" is a subset of a topological space which is equal to its own derived subset, i.e., it is a closed subset without isolated points. Note that this is a property of subsets rather than spaces.

Strangely, "perfect set" still redirects to "perfect space", and "perfect space" does not mention perfect sets at all. It that terminology obsolete? Boris Tsirelson (talk) 08:30, 18 August 2013 (UTC)

Opinions
A quote from talk:WikiProject_Mathematics of 21 November 2013 (Section "Perfect set versus perfect sets"): --- Perfect set redirects to Perfect space, while Perfect sets redirects to Derived set (mathematics). Rather strange. Boris Tsirelson (talk) 08:34, 20 November 2013 (UTC)
 * The first redirect should go in the other direction. Perfect sets (contained in some larger topological space) are the more important notion; just because the concept can be defined intrinsically doesn't mean it should be. --Trovatore (talk) 08:39, 20 November 2013 (UTC)
 * I see. And what about "Perfect sets"? Why such entry? Boris Tsirelson (talk) 11:19, 20 November 2013 (UTC)
 * That should also redirect to perfect set, of course. --Trovatore (talk) 17:31, 20 November 2013 (UTC)

--- Boris Tsirelson (talk) 20:47, 21 November 2013 (UTC)

Move and rewrite slightly

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section. 

The result of the move request was: 'Closed as move. This page was moved during the move request, which should not have occurred per WP:NOMOVE. However, given that the listing period is now over and there are effectively three supports with no opposes, it seems fine to keep the new name. &mdash; Amakuru (talk) 14:17, 2 December 2013 (UTC)

Perfect space → perfect set – In my experience, the most usual definition one comes across for this notion is, S is perfect if S is closed and has no isolated points. Now, of course, every topological space is closed in itself, so if you insist on treating the notion as an intrinsic one about a whole space, instead of as a property of subsets of a space, then the "closed" part trivializes.

But that trivialization leads to pathologies. For example, the rational numbers with their usual topology are a "perfect space" in that sense (or a perfect subset of themselves, in what I argue is the more usual nomenclature). But that sort of perfect-ness has no connection with the reasons that perfect sets have historically been considered important.

The historical context for the notion is things like the Cantor–Bendixson theorem, which shows in particular that closed sets (of reals) cannot be counterexamples to the continuum hypothesis, because they can be decomposed uniquely into a countable set and a perfect set, and a nonempty perfect set must have the cardinality of the continuum. That's completely lost for pathological examples such as the rationals-in-themselves.

More modernly, perfect sets are important as the sets of branches of perfect trees, which are trees that always branch past any given node. Again, that has little connection with this intrinsic "perfect space" notion.

So my proposal is: Move the present article to perfect set, and reword the introduction to frame it as a property of subsets of a given topological space (closed and having no isolated points). Then mention the notion of perfect space, which is simply a space that is perfect as a subset of itself, and point out that this is the same as having no isolated points. Trovatore (talk) 19:44, 22 November 2013 (UTC)


 * I support the move. Perfect space is not enough notable for a separate article. Boris Tsirelson (talk) 15:05, 23 November 2013 (UTC)

I have rashly gone ahead and moved it. I haven't yet re-written the introduction, nor dealt with links to the new redirect page. Michael Hardy (talk) 22:23, 23 November 2013 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.