Talk:Separable space

Merge
This needs to be merged with separable space. Charles Matthews

(This has been done.)

Condense
Hi Fropuff,

I do not think your condensing made the article clearer. You removed my introduction sentence where I tried to give an intuitive explanation for the laymen (or not topologist) in terms of approximation. Furthermore although the ==Definition== section was quite short it was nonetheless useful for the reader as he could immediatly jump the the definition of the concept without scanning the whole article.

Your other edits, especially on the properties I added, are an improvement and made the article clearer.MathMartin 12:21, 6 Mar 2005 (UTC)

Hmm, User:Charles Matthews's edits made my points above somewhat moot. I still would like to see the definition separated out into a ==Definition== section but now I can live with the page. MathMartin 13:20, 6 Mar 2005 (UTC)

Sorry, I hadn't really looked at what changes you'd made before I made mine. In most cases I would agree with you regarding the definition section, however, in this case I think the definition is short enough and clear enough that it doesn't hurt to include it in the first sentence. -- Fropuff 15:36, 2005 Mar 6 (UTC)

Name
Why "separable"? What's being separated? Is it something like, e.g., a rational number separates any two real numbers? —Ben FrantzDale 23:42, 25 October 2006 (UTC)

Properties
These two properties are a bit unclear:


 * Every second-countable space is separable.
 * A metric space is separable if and only if it is second-countable and if and only if it is Lindelöf.

Clearly a MS is a space so if separable iff second countable; so should the second one not be

In a Metric space the following are equivalent: -- space 2nd countable -- space separable -- space Lindelof

This is not an edit as I would need to check first in Kelley.

Alanb 11:23, 18 April 2007 (UTC)

"non-separable Hilbert spaces in theoretical physics" - TWO COMMENTS.
This concerns the statement: "The possible use of non-separable Hilbert spaces in theoretical physics has provoked some inconclusive debate."

(1) Can someone expand further on this cryptic comment, perhaps citing a reference where one can learn more?

(2) I was a little disturbed to find that Google knows about just two sites that have the phrase "non-separable Hilbert spaces in theoretical physics": this site (this wikipedia page) and the following .com site:

http://www.nationmaster.com/encyclopedia/Separable-space

Who has borrowed this intellectual content from whom? I suppose in the end there is nothing to be done...

DrTLesterThomas (talk) 23:07, 9 June 2008 (UTC)

Mistake?
"The product topology on the set of all functions (not necessarily continuous) from the real line to itself is a separable Hausdorff space. " is this true? It is obvious that a separeble Hausdorff space has cardinality <= IN^IN. This space has cardinality c^c "A separable, Hausdorff space X has cardinality less than or equal to 2^c" why is IN^IN >= 2^c = c^c? It looks like that c is the cardinality of the reals. If c would be the cardinality of the natural numbers, I think it would be OK —Preceding unsigned comment added by 77.4.18.100 (talk) 09:01, 4 July 2008 (UTC)

I can't say whether or not it is "obvious" that a separable Hausdorff space has cardinality at most $$\aleph_0^{\aleph_0} (= c)$$, but I can tell you that it's not true. All the assertions made on this page about separability and cardinality are true. What is needed is not verification but references. Plclark (talk) 17:50, 6 July 2008 (UTC)Plclark

Why is it not true? Take a countable dense subset A, then there is a surjection A^IN -> X, sending a series to its limit (here, we use Hausdorff) or, if it does not converge, to a fixed point. So |X| <= IN^IN. —Preceding unsigned comment added by 77.4.188.4 (talk) 17:44, 7 July 2008 (UTC)


 * Munkres, 1ed, exercise 13, p195, asks one to show that [0,1]^[0,1] has a countable dense subset in the product topology. JackSchmidt (talk) 18:37, 7 July 2008 (UTC)
 * Willard, 1ed, theorem 16.4c, p109, says that a product of Hausdorff spaces, each with at least two points, is separable if and only if each factor is separable and there are at most c factors. Here separable means countable dense subset, and c is the cardinality of the continuum. JackSchmidt (talk) 18:45, 7 July 2008 (UTC)
 * OK, and what is wrong with my argument? —Preceding unsigned comment added by 77.4.180.34 (talk) 23:33, 7 July 2008 (UTC)

Your argument is valid if the space is first countable. In an arbitrary space it need not be the case that every element in the closure of a subset is the limit of a sequence of elements from that subset. For more details see sequential space and Net (mathematics). Plclark (talk) 01:34, 8 July 2008 (UTC)Plclark
 * Can you give a nice Hausdorff, separable example of this? I think my examples all fail to be Hausdorff. Metric spaces are always first countable, right? JackSchmidt (talk) 14:47, 8 July 2008 (UTC)

I'm not sure what you're referring to in the first sentence. The space $$[0,1]^{[0,1]} $$ is Hausdorff, separable, not first countable, of cardinality $$2^c$$. References for all these facts should be available in Willard's book and/or Engelking's book. Plclark (talk) 00:53, 9 July 2008 (UTC)Plclark
 * "Can you give an example of an element in the closure of a set that is not the limit of a sequence from the set?" JackSchmidt (talk) 02:29, 9 July 2008 (UTC)

Did you read the article on sequential spaces? It gives the example of the cocountable topology on an uncountable set. Another example is given in the article on first countability...I would like to say this politely but I'm having trouble, so please forgive me and pretend that I found some polite way to say it: it seems to me that the purpose of a talk section is to discuss the merits of possible edits to the article, not to give tutorials on the subject matter of the article. If you want to edit this article, maybe you should consult a book on general topology to help you do so. Plclark (talk) 07:21, 9 July 2008 (UTC)Plclark
 * I think this response is quite rude. It is not in keeping with wikipedia's policies, nor is it mathematically correct.  Wikipedia articles are intended to be edited by everyone; you may be interested in the hundreds of WikiProjects that exist to help cleanup articles like this one. The cocountable topology on an uncountable set is not Hausdorff.  I think your response is quite uncivil and would appreciate a retraction and apology. JackSchmidt (talk) 13:20, 9 July 2008 (UTC)

Here is a good example, the Arens-Fort space. It is in Kelley p77 problem E, and Steen and Seebach p54, example 26. This is Hausdorff, countable, separable, and the countable subset X-{(0,0)} has (0,0) as a limit point, but no sequence from X-{(0,0)} has (0,0) as a limit. The space looks like a series of fixes of the particular point topology in order to get Hausdorff. Kelley states the result fairly clearly, but S and S has the proofs. JackSchmidt (talk) 18:54, 9 July 2008 (UTC)

You rephrased the question as "Can you give an example of an element in the closure of a set that is not the limit of a sequence from the set?" I thought you wanted a general example of this phenomenon and didn't realize that you also wanted separable and Hausdorff. So I thought you had not looked at the wikipedia references I gave and got a little frustrated. There seems to have been a miscommunication; I'm sorry for that. (Notice that my first apology came in advance; I will even apologize again further below...wait for it.) Of course then the space $$[0,1]^{[0,1]}$$ satisfies all the requirements of your question, and the answer is given by the countable dense subset which exhibits its separability.

I agree that everyone can edit any articles they choose, and I didn't tell you not to edit this (or any) article. I said that if you want to edit it but have questions about the content of the material, you should consult a reference which answers those questions, rather than post inquiries about them on the talk page. Evidently there's a continuum between questions/concerns about the article itself and asking questions about the material on which the article is based...in this case I thought that the line was getting toed, at least. It seems that my response provoked you into doing exactly what I asked -- consulting general topology textbooks -- and I'm sure you'll not argue with me that consulting general topology textbooks is a perfectly good thing for anyone to do at any time and doubly so if they are editing an article on general topology. I wish I had managed to say something which engendered that part of the reaction but didn't bother and/or offend you. I'm sincerely sorry for that; it was not my intention. Plclark (talk) 20:22, 9 July 2008 (UTC)Plclark
 * Thanks for the apology; a miscommunication is obviously a problem on both ends, so my apologies as well for not understanding your intention. I did read the articles you mentioned first (sequential space and net), but they either didn't have an example, or had a not-hausdorff one.  I had even checked the books on hand, and didn't see anything.
 * The question was intended to improve the article. The anon raised a legitimate concern, and it is fairly easy for a subject expert to address this concern and improve the article.  Presumably this could be done by giving an example, with a nice reliable source.  Experts are busy, and often don't have sources at hand since they know the material well enough to not need them every moment.  A great variety of people work on the article, and it is not necessary for them to understand the material (at all) for them to help with this situation.
 * I was asking you to explain the material a bit, so I could format it into the article. A fairly explicit example was necessary so that it could be easily sourced.  Once the example was patiently explained to me, it was easy for me to find sources.  I think my plan right now is to make the A-F example very clear on the A-F page, and then it will be easy to link from here.  Similarly, the R^R example should probably be made clear on a page of its own, and then it will be easier to link here.  The examples section should be fairly easy reading (no detailed justifications, like it currently has), so using examples with wiki articles already written will be helpful.
 * Just to make it clear, since I think this is non-obvious: there is no reason to think I should be able to understand general topology in order to improve this article. I came here solely because this article recently made it on the list of "failed verifications" articles (an article where a claim is made, then a citation to a reliable source is made to justify that claim, but then someone else disagrees that the source actually justifies it).  It is a matter of official policy (WP:V) that more or less any fool can resolve such disputes.  Now as it happens, the editor had just used the wrong cleanup tag, and instead what was needed was sources.  This is a little harder, but really not very much.  All I've done is try to find sources for the claims in the text; this basically requires a library and the ability to read an index.  Assuming that I understand all that mumbo jumbo in the books is probably entirely too generous.
 * I mention this because we definitely have extremely (extremely) productive, useful, good, etc. editors who may not know beans about certain (or most) areas of math. It is very important to try and help them (or at least politely decline).   The subject expert is a desperately needed ingredient, but there are lots of other necessary parts too. JackSchmidt (talk) 21:37, 9 July 2008 (UTC)

Just to point out a few things (unrelated to this argument), there are a few nice examples of spaces that are Hausdorff, separable but do not have the property that a point in the closure of a set is the limit of a sequence of points belonging to that set. For example, consider Rω in the box topology. This space has a countable dense subset and is Hausdorff but does not satisfy the desired condition. A proof is as follows:

Take the set X = {(xi) | xi > 0 for all i}; note that 0 is a limit point of this set. Suppose there is a sequence of points in X converging to 0. Let (sn) be that sequence and let the nth term, sn = (s1n, s2n, s3n…..). Then choose the basis element about 0 to be:

B = (-s11, s11) X ……X (-snn, snn)…..

Then B cannot contain any term of the sequence (sn) since the nth co-ordinate of this term doesn’t belong to (-snn, snn).

Q.E.D

Actually, if a space is first countable, then if a point belongs to the closure of a particular set, it must be the limit of a sequence of points in that set.

Topology Expert (talk) 04:25, 10 July 2008 (UTC)

More comments
I have now read the article in more detail; here are some further comments. (On pages that others are involved in editing, I prefer to suggest changes / problems on the comment page first rather than to make the edits myself. I will take a lack of response as approval to make changes.)

1) The lead sentence says

"In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space."

So far as I know, in all areas of mathematics a topological space is called separable if it contains a countable dense subset. I propose simplifying to "In mathematics..."


 * The reason that topology is mentioned here is mainly to give context, not to limit scope. Indeed, many openers in WP give context in this style (and I'm not sure if I like it myself). But regardless, in algebra there is quite a different notion of separability. See, for example, separable polynomial. Also, in PDE/ODE, there is a notion of a separable equation. So restricting to topology here is a good idea. Oded (talk) 16:25, 9 July 2008 (UTC)


 * One other weird part is that "separable topological space" even had multiple meanings in topology according to Kelley's 1955 General Topology. Willard (1970) and Munkres (1975) don't mention the problem, and neither does Sierpinski (1952), so it isn't clear to me how widespread the conflicting notation was.  There is an article by Birkhoff 1933 that seems to use "locally separable" for first countable, and "perfectly separable" for second countable, but does not seem to use "separable" itself for anything. JackSchmidt (talk) 16:54, 9 July 2008 (UTC)


 * Yes, as a mathematician I find it slightly strange to start mathematics articles with In mathematics..., but if that's the convention here so be it. And, sure, of course the word "separable" means many other things in other areas of mathematics: but none of these usages are applied to topological space.  At least not recently: I looked back at Kelley's book after JackSchmidt's comment, and Kelley says that some people use separable to mean "second countable".  This makes perfect sense in metrizable spaces (since the concepts are equivalent) but in general topological spaces is clearly incompatible with the rest of the standard nomenclature.  As you say, Kelley's book was written in 1955; 50+ years is certainly long enough for terminology to change.


 * Anyway, I suggest the lead sentence begin "In mathematics, a topological space is called separable if..." together with adding some disambiguation about other mathematical uses of separable. Plclark (talk) 20:36, 9 July 2008 (UTC)Plclark
 * Since topological space pretty quickly directs the reader to topology, I think having both in the first sentence is not needed. So "In mathematics, a topological space is called separable if..." sounds good to me.  Setting the context usually takes two steps, first that it is math, then that it topology, algebra, analysis, or whatever.  One way to get used to this idea is to visit a RandomPage, or look at some of the NewPages.  For the latter especially, it becomes clear that an article could be about *anything*.  JackSchmidt (talk) 20:54, 9 July 2008 (UTC)


 * I don't object to any of your suggestions for the lead sentence. Oded (talk) 22:39, 9 July 2008 (UTC)

2) "the same way that any real number can be approximated to any specified accuracy by rational numbers, a separable space has some countable subset with which all its elements can be approached, in the sense of a mathematical limit."

I don't completely agree with this sentence, since "approximation" connotes to me an essentially metric notion. The sense in which an arbitrary point in a separable topological space is "approximated" by a countable dense subset seems obscure. Also the language seems to suggest to me the limit of a sequence and as above this is not generaly valid. I suggest deleting this sentence entirely.

3) "Separable spaces are topological spaces with a certain limitation on their size." The exact meaning of this is obscure to me.  Better would be something like saying (not necessarily at this point in the article) that there are no cardinality restrictions on an arbitrary separable space but separable spaces which are either first countable or Hausdorff have certain cardinality limitations.

4) I am on the fence as to whether the discussion of the historical rise and fall in stature of separable spaces is interesting and deserves to be elaborated upon or is on the trivial side and could be omitted. But I propose that there should be either more or less (!) information on the topic than there is now.  Dieudonné should be referenced.  The sentence about theoretical physics is vague and also a bit weaselly.

5) The part about using separability for algorithmic proofs in constructive mathematics is definitely of interest: references please.

6) In the examples section, also $$\mathbb{R}^{\aleph_0}$$ is separable in the product topology, which is equally important and arguably more interesting.

7) "The space of continuous functions on the unit interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (this is the Weierstrass approximation theorem)."

The Weierstrass approximation theorem says that the polynomials with real coefficients are uniformly dense in [0,1]. Note that this is an uncountable set so does not demonstrate separability. This is easily fixed, since the polynomials with rational coefficients are seen to be uniformly dense in the polynomials with real coefficients. Also there are more purely topological proofs of the separability of certain function spaces, I believe; a reference to one of them would be nice.

8) "It can be shown that any separable Banach space is isometrically isomorphic to a closed linear subspace of this space, by the Banach-Mazur theorem." That this is the Banach-Mazur theorem is obscure here.  Better: "According to the Banach-Mazur theorem, any separable...."

9) "The trivial topology on any set is separable since any singleton is dense. This shows that by removing the Hausdorff requirement in the previous example we can get separable spaces with arbitrarily large cardinality." Wasn't this already shown above, using the particular point topology?

10) "The continuous image of a separable space is separable, (Willard 1970, Th. 16.4a). It follows that separability is a topological property preserved by homeomorphisms." This is a strange deduction.  That separability is preserved by homeomorphisms follows immediately from the definition.

11) "Every product of at most c Hausdorff, seperable spaces is separable, (Willard 1970, Th 16.4c)." Note that this justifies that $$[0,1]^{[0,1]}$$ is separable, which was questioned above!  Reorganization seems necessary.  (There is a spelling mistake here; I'll go ahead and fix that right away.)  Plclark (talk) 01:24, 9 July 2008 (UTC)Plclark

Rewrite
The rewrite was a little drastic, but overall a huge improvement.

In particular, the prose sections are a huge improvement over bulleted lists of poorly sourced material. Splitting the examples into early and further also seems like a huge improvement. I think also the choice of the division is wise: even though it is probably more important that the function spaces are separable, I think some of the main reasons for this are that their separability means they are at least vaguely similar to the very familiar examples given first. One might even try to say this explicitly for the separable Hilbert space, as sort of an infinite analogue of R^n.

I added wikilinks and corrected basic WP:MOS problems. Only one should be avoided in the future: try to leave each paragraph as a single line so that changes are visible.

There are still some problems left to fix:
 * Once the content has settled, the lead needs to be written. (probably a little more drastic rewrite near the bottom first)
 * There are still lots of missing wikilinks. I added most that I knew for topology, but there are just random ones too.  I figure anything topology wise is relevant, but I didn't want to overlink (polynomial ring etc.).  Limits, continuous image, and some related guys are unlinked, since I didn't know which article was best.
 * One of the (further) examples got chopped. I think the line can be deleted, but it said "The product topology on the set of all functions (not necessarily continuous) from the"
 * The use of "we", "easily", "clearly", and blackboard bold are discouraged by the math manual of style, WP:MSM
 * The use of inline TeX is somewhat discouraged

The first is easy once the rewrite is done. The second should be easy for Oded or other people more familiar with the topology articles. The third seems clear, but I wanted to make sure Plclark didn't have a plan for it. The fourth can be fixed easily, especially if everyone is happy with the basic language. The fifth is somewhat a matter of style, but basically wikipedia's tex engine produces extremely jagged prose paragraphs when this is used, so it should be avoided if unicode and wiki markup will suffice. JackSchmidt (talk) 02:33, 10 July 2008 (UTC)

(edit conflict) Congrats on the nice rewrite! There is one issue that is not really clear to me. That is the upper bound proof of cardinality. First, the definition


 * $$F(x) = \bigcup_{i_0 \in I} \{x(i) \ | \ i \geq i_0 \}$$

Shouldn't that be


 * $$F(x) = \Bigl\{\{x(i) \ | \ i \geq i_0 \} : i_0\in I\Bigr\}$$

instead? Also, this whole passage requires a bit more explanation. I know what a filter is. But what is a "filter base of tails"? Where is that defined? Also, is there a phrase missing in that sentence? Otherwise, it seems that there is a filter associated with a filter. Oded (talk) 04:09, 10 July 2008 (UTC)
 * Also, one eventually wants a published reliable source for the proof. Will Kelley have such a proof?  Chapter 2 is on nets, I think.  Willard is good for refs, since it has a chapter or two on separable spaces.  I think Munkres will be useless since it doesn't have much on separable spaces.  Anyone want to search Zbl for some early foundational papers in the area?  I think Sierpinski cites a few, but I don't know how to judge the papers or the completeness of the reviews).  I added textbooks to the references.  I think most things in the article are covered in at least one of them, but usually not *most* of them, so inline cites are probably appropriate. JackSchmidt (talk) 04:17, 10 July 2008 (UTC)

To Oded: yes, the formula should be as you wrote: thanks for catching that (and texing it up nicely to make it easy for me to change it). I also changed the first "associate" to "define", which therefore (at least in some sense) answers your question: it is defined right there! And then I am associating a filter to a filter base, which is a standard thing to do and should be covered in the filter article (you just throw in all supersets of what you have already). As for a reference? In the unofficial sense (i.e., for you to read up on) I recommend http://www.math.uga.edu/~pete/convergence.pdf, pages 23-26. But I certainly do not want this to be linked from wikipedia: these are just unofficial notes and have not been refereed or anything like that. They do include (somewhat sloppily, but they're there) real references though, so I believe an appropriate reference for this back-and-forth between filters and nets would be the 1957 Monthly article "Filters and equivalent nets" by M.F. Smiley.

Some references are still needed (the more the better, I say). In my experience Willard's book is the best single reference. For no good reason I don't own a copy, though, so I did not take the proof from there but rather came up with something on my own. That is an issue to be discussed, I suppose: I don't think the stigma against "original research" should apply to brief arguments that a qualified writer puts into an article, especially if they are accompanied by a reference to a (not necessarily identical) proof in a reputable source. But what do people think about this? (To be honest, I put in the proof to liven things up a little in an otherwise somewhat soporific article.)

The only significant passage I took out was the bit about separability going in and then out of favor together with a dropping of Dieudonnè's name. It just seemed uninformative to me. Also isolating the "constructive" aspects of separability into its own section highlights how incomplete the discussion is. It would be nice to see more about this, but that's way outside of my area of expertise.

Going a little bit against what I said before, I remembered that "separability" is also used in a topological vector space to mean that it has a countable topological basis. I believe that this indeed happens iff it is separable as a topological space, but this should probably be mentioned somewhere. (Strangely, topological vector space says nothing about separability.)

There is now little to nothing about the "history" of separable spaces. I'm not sure how much there is to be said about this. Plclark (talk) 05:50, 10 July 2008 (UTC)Plclark

About the cardinality proof
Regarding Prof. Schramm's change's to the cardinality proof: I think that the proof I wrote was a bit more informative, because the given characterization about closure in terms of filter bases, while correct, is different from any of the standard characterizations I have seen in the literature (e.g. this characterization does not appear on filter (mathematics) or the notes on convergence that I linked to above.  In fact what was deleted in my version of the proof was precisely one way to show that this characterization holds.  However, my proof is also not ideal because it relies on the back-and-forth between nets and filters.

So any way around I think more information is necessary, probably in other articles besides this one. Perhaps a new article is needed on nets vs. filters, or maybe this material should go in the (far from satisfactory) article Modes of convergence?

In the meantime, if one of us can find a reference for the aforementioned characterization in terms of filter bases, that would be helpful. Plclark (talk) 22:54, 10 July 2008 (UTC)Plclark


 * Unless I am thoroughly confused, it does appear in the page filter (mathematics) in the section filter (mathematics). As you say, you have given a sketch of the proof the convergence of nets implies convergence of filter bases, but omitted the definition of what it means for a filter base to converge. In fact, it is likely that at most of the references where convergence of filter bases is defined, you will also have the statement that the closure of a set is characterized by them (as is indeed the case for the WP article). I tend to agree with you that convergence of nets is better known, but it is actually not so helpful here. It is just as easy to see that the closure is characterized by convergence of filter bases as it is to show that if there is a net that converges then the induced filter base also converges. Somehow, convergence of filter bases is more natural, as it does not use anything completely exterior to the space (such as a directed set).
 * I now made it even more explicit in the filter (mathematics) article. Oded (talk) 23:49, 10 July 2008 (UTC)
 * By the way, I was hesitating if it is worthwhile to include in the filters article the application that the closure of a set Y in a Hausdorff space has cardinality at most $$2^{2^{|Y|}}$$. Oded (talk) 23:34, 10 July 2008 (UTC)
 * (Reply to "by the way") I had wondered if there was some bound on how big  a closure could be. Can it be shrunk further in a metric space?  I think it would be reasonable to include such an application somewhere (filters seems good), and link casually from Hausdorff space and here. JackSchmidt (talk) 02:26, 11 July 2008 (UTC)


 * I have already added this remark in the current article. Putting it also in the articles on filters and/or closure might be a good idea.  Plclark (talk) 08:56, 11 July 2008 (UTC)Plclark

There may possibly be a little confusion on my part and/or yours. A possible source is the ambiguity inherent in the term "limit point". I myself use the term to mean "cluster point" but some others use it to mean "a point of convergence (possibly nonunique if the space is not Hausdorff) of the filter." I believe the most standard characterizations of closure in terms of filters are the following statements:

1) A point x is in the closure of a subset Y of X iff there exists a filter base F on X which has X as a cluster point -- i.e., every element of F intersects every neighborhood of x.

[The filter base which exhibits the proof is just F = {Y}. Evidently this does not converge to x in general.]

2) A point x is in the closure of a subset Y of X iff there exists a filter F on X which contains Y as an element and converges to x.

[Here we show that the filter bases {Y} and N_x are "compatible" in that there exists a filter containing both of them. Note that any such filter contains all neighborhoods of x in X and therefore has no cardinality restriction in terms of Y.]

Note that both of these characterizations are essentially immediate unpackings of the terminology. In contrast, the characterization we want seems to me to lie a bit deeper, so at the least should get an inline citation and perhaps a proof sketch.

But it is time for me to consult Willard's book -- with luck, I can check it out of the library tonight. More later. Plclark (talk) 00:54, 11 July 2008 (UTC)Plclark


 * The filter article defines "limit point" of bases. How do we show that if $$x\in\bar Y$$ then there is a filter of subsets of Y converging to x? This is really easy and nicer than the corresponding statement for nets. The filter just consists of all the intersections $$U\cap Y$$ where $$U$$ runs over all the open sets containing x. This is clearly the simplest explanation of the fact that limits of filter bases characterize closure (and not the one using nets). Oded (talk) 01:40, 11 July 2008 (UTC)

OK! I agree that this is correct and is simpler than factoring the problem through nets. However this fact and its proof, simple though they are, does not seem to be found in any of the standard references. I do now have Willard's book in my possession and it does not have a proof of this, nor of the consequence that we are concerned, with, namely cardinality bounds on separable spaces. In fact this seems to be a low-point of the exposition in Willard's otherwise excellent book, because he gives as Exercise 16F1 that a separable first countable space has at most continuum cardinality, whereas anyone who has read the current wikipedia article knows that a sufficiently large set endowed with the trivial topology gives a counterexample to this: Hausdorff is needed to get any cardinality bounds. (And he does not mention the case of Hausdorff but not first countable.)

Perhaps the discussion should now be switched to filter (mathematics): I feel that Section 2.3.2 is not yet satisfactory. None of the five bulleted assertions has a proof or an inline citation, and it seems unreasonable to ask a reader to look through multiple books including Bourbaki to find them! (I myself often prefer including quick and easy proofs, if this can be done so as not to interrupt the flow of the article.) As an instance of the problems that this induces, note that the fourth bulleted assertion seems to be incorrect: take e.g. X = {p}. (I conjecture that this is an instance of the limit / limit point / cluster point / omega-accumulation point confusion.) Plclark (talk) 02:31, 11 July 2008 (UTC)Plclark

Wait, I'm sorry -- the fourth bulleted assertion is using the definition of limit point given in the third bulleted assumption: what I would call an accumulation point. With this definition the fourth bulleted assertion is correct. But it seems unnecessarily confusing to me to use "limit point" for both "point to which a filter base converges" and "accumulation point of a subset." A terminology change may be in order. Plclark (talk) 03:05, 11 July 2008 (UTC)Plclark

Generalization of separability?
According to Willard, the property "having a dense subset of cardinality $${\aleph}$$" is a topological property. Thus separability is only a special case of this more general topological property.

I don't know if "having a dense subset of cardinality $${\aleph}$$" for any $${\aleph}$$ other than $${\aleph_0}$$ is "interesting", but perhaps the more general notion could be mentioned. I would personally be happy to see a non-trivial example that is $${\aleph_1}$$-separable but not $${\aleph_0}$$-separable (= separable). Come to think of it, the Stone-Chech compactification should be able to provide a few examples of $${\aleph}$$-separable spaces.

EDIT: There is sort of a mention of this generalizatin in the section "Cardinality". YohanN7 (talk) 16:56, 1 August 2009 (UTC)
 * The generalization is usually treated as a cardinal function called density. Ntsimp (talk) 14:45, 31 August 2011 (UTC)

Cardinality bound
Hey guys I found an excellent new proof of the cardinality bound for hausdorff separable with no filter crap.

Let D be a countable dense subset of space X We'll define an injection F from space X to the set 2^P(D) which has power continuum cardinality. For any point x of X, define the A'th coordinate of F(x) (where A is a subset of D and thus is a coordinate of the tuple F(x) in 2^P(D)) to be 1 iff there exist a neighborhood U of x such that

A=(D intersection U)

It's easy to check this is an injection once you get your head around the confusing terminology. Seemed to me it might go through for T1 but a closer inspection showed no. Sorry I dunno how to use wiki. Najor Melson (talk) 06:36, 6 November 2009 (UTC)
 * The concept of a filter is one of central importance and interest in mathematics (please do not swear in the manner that you have done - it is insulting to some). Nevertheless, your proof appears to be correct (and I think the most common manner in which one may prove this fact). Below, I have typed your proof in LaTeX, for readability. For most simple commands, LaTeX is rather intuitive and not hard to use (when you edit this page, you can look at how the LaTeX is done):


 * Let $$X$$ be a separable space and let $$D$$ be a countable dense subset of $$X$$. We will define an injection, $$F : X \to 2^\mathcal{{P}(D)}$$, as follows. If $$x$$ is an element of $$X$$, and $$A$$ is a member of $$\mathcal{P}(D)$$, define $$F_A\,{(x)} = 1$$ iff $$A = {D}\cap{U}$$, for some neighbourhood $$U$$ of $$x$$, and 0 otherwise. The function $$F(x) = \{F_A\,{(x)} \}_{A \in \mathcal{{P}(D)}}$$ is the desired injection. -- PS T  07:33, 7 November 2009 (UTC)


 * I know filters are very important. IMO the best looking proof of Tychonoff's theorem is using ultrafilters. But after reading the sections above me I got the feeling that people don't like the filter proof for cardinality bound. The proof is from Counterexamples in topology page 26; there's also another pretty cool theorem above it - compact hausdorff topologies are minimal hausdorff and maximal compact, so they are very neat and "right in the middle of the lattice of topologies". Sorry if I insulted you Najor Melson (talk) 11:19, 7 November 2009 (UTC)