Talk:Subbase

Without heading

To explain my taking out "which covers X"...this may seem wrong at first, because you have to get the whole space X as an open set. The way you get it is with empty intersections. When you take the empty (in particular) finite intersection, i.e. the intersection of the empty subcollection of the subbasis, you get X. A good explanation of this is at nullary intersection. So, the worst you can possibly do, if the subbase is empty, is get the indiscrete topology. I hope this clears things up a bit. Thinking in terms of finite intersections forming a base is fine if you always remember to throw in X. This is a kind of "constructive" way of getting the topology. I prefer to think of it abstractly (even _define_ it abstractly) as the smallest topology containing the collection. This way, "generates" has the same meaning for base and subbase, it just means that the topology generated by an arbitrary collection (which might not be a base) might not be expressible as the set of all unions of elements of the collection. Using the constructive definition is good to play with examples, the abstract definnition is good to prove things. Either way, once you define it, you can prove the defs are equivalent.

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There are a number of things wrong with this page, and I will be correcting them.

There are authors who like nullary intersections, there are authors who do not. WP should not take a position on this. On the other hand, it is too trivial to play it up.

It is silly to claim that finding an instance of a definition is a use of the definition. It is a "use" in the pedantic sense that you took the definition and worked out what it meant, but it is not a mathematical use in that you are getting something back for your investment in this concept.

The real uses are two. Certain topologies (weak, product) are quite naturally defined in terms of subbases. And the Alexander subbase theorem, which I'll be adding with a proof sketch.--192.35.35.36 19:28, 24 Feb 2005 (UTC)

I don't understand what you mean by "it is silly..." When I said, "using the constructive definition", I meant e.g. by way of defining weak or product topologies by subbases. Here, it's good to have a concrete idea that the basis elements look like finite intersections of subbasis elements. When I said, "using the abstract definition", I meant there are certain proofs which follow more naturally by using the "smallest topology containing" definition. I consider each of these cases to be "getting something back for your investment". Revolver

Can you give an example of authors who "don't like nullary intersections"? To my understanding, they may emphasise the case of the nullary intersection for clarity, but I don't recall any that say it's incorrect to use the nullary intersection. In any case, I think you missed my point. Say you have a subbasis which covers X. Fine, you say the subbasis "generates" X, in the sense that the generated topology is the smallest containing the subbasis. But, if the subbasis does not cover X, then the "constructive" definition (unions of finite intersection, without the convention of nullary intersection) does not give the topology generated by X. Only the "abstract definition" gives the right topology. So, throwing out the nullary intersection negates a whole range of examples which "make sense" in the "abstract definition". To me, this is the most important reason for including nullary intersection in this case. You either have to include nullary intersections, or stipulate that the collection is a cover in ad hoc fashion. Either way solves the issues, just the former makes much more sense to me. Revolver

an example

In topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T such that the collection of finite intersections of elements of B form a basis for T. This means that every open set in T can be written as a union of finite intersections of elements of B. Equivalently, this means that given x∈U with U open, there are finitely many members S1, … ,Sn∈B such that x∈S1∩…∩Sn⊆U. We say that the subbase generates the topology T, and that T is generated by B.

No, here is where I disagree. Consider Sierpinski space. I say (and I believe this is shared) that if Sierpinski space = {0, 1}, where, 0, {0}, and {0, 1} are open, then { {0} } generates this topology, i.e. the Sierpinski space is the smallest topology containing { {0} }. In other words, { {0} } should be considered a true subbasis of Sierpinski space. However, under your definition above, "equivalently, given x in U with U open,..." { {0} } is not a subbasis, because, e.g. 1 is in the open set {0, 1}, yet there do not exist finitely many members S1,..., Sn of { {0} } such that 1 is in their intersection, which is contained in {0, 1}. (Because, of course, B only has one element, namely {0}, and 1 is not an element of {0}.) This is what I mean by the difference between the constructive and abstract definition. In the abstract definition, you don't have to puzzle over nullary intersections, in the constructive definition, you have to either allow them or explicitly throw in the whole space. Incidentally, I was looking at Joshi's book, and he presents it confusingly, too. It's a minor point, but we were quibbling over minor points. Revolver

Your statement becomes true if you put the word "proper" in front of "open set". Which is more inelegant: specifically excluding the whole space from this condition that must be met, or specifically including the nullary intersection? It seems a matter of taste, I guess, but the latter seems to me the far more elegant of the 2 "inelegant" alternatives.

Just to be clear, the problem I see isn't so much with the particular case of the indiscrete topology. The point in allowing nullary intersections isn't just so that we can elegantly "define" the empty subcollection to be a subbasis for the indiscrete topology. It's so that the two definitions ("constructive" and "abstract" above) agree. Without allowing nullary intersections or stipulating that it's a cover in an ad hoc fashion, these 2 senses of "subbase" will disagree whenever the subbase doesn't cover X. Adding the stipulation that the collection must cover X is unacceptable to me, because it creates a division between the 2 definitions of "subbase". To give an example, with this stipulation, statements such as "every family of subsets of a set X is a subbase for a unique topology on X" become simply false.Revolver

Forms a basis only when it's with the Empty set???

Wiki says..

The collection of open sets containing the empty set, X, and all finite intersections of elements of B forms a basis for T.

And it seems to mean...

{empry set} U {X} U {all finite intersections of elements of B} is a basis for T.

From the less wordy one I wrote above, Is {empty set} really necessary?

If so, it would be useful to explain in the article why it is necessary.

Cosfly 07:09, 2 February 2007 (UTC)

Subbase generated by (-infty,a)

The subbase consisting of all semi-infinite open intervals of the form (−∞,a) alone, where a is a real number, does not generate the usual topology. The resulting topology does not satisfy the T1 separation axiom, since all open sets have a non-empty intersection.

I don't follow this: finite intersections of two of these guys yield (min(a,b),max(a,b)) and (-infty,a). Is it saying that every union of these guys will be nondisjoint from every other union? That's patently not true. —Preceding unsigned comment added by 24.57.117.130 (talk) 18:21, 2 November 2007 (UTC)

A late reply, but that line is still in the article. Unless I am mistaking, it seems to me that

$\bigcap _{i}(-\infty ,a_{i})=(-\infty ,\inf _{i}a_{i})$

$\bigcup _{i}(-\infty ,a_{i})=(-\infty ,\sup _{i}a_{i})$

In particular taking finite intersections and arbitrary unions, we end up in the subbase or the empty set or R. So I think this collection of intervals (-infty,a), together with empty and R, is already a topology. If this is true, then it needs no comment that it does not yield the usual topology (e.g. it does not contain any bounded interval). And indeed it isn't T1: a singleton {x} is not closed, as R-{x} is not open, not being an interval. 145.97.197.215 (talk) 15:50, 18 August 2011 (UTC)

Has anyone seen "semibase" used as a synonym of subbase?

I've just encountered the term "semibase" instead of subbase, here: b:Topology/Bases#Semibases. (The original formulation was a bit unclear; but, on the other hand, it is clear from the history, that the author first intended to use the term "subbase", but changed their mind. This, I think, strongly suggests, that indeed the same concept was intended.)

Now, I personnally think the term "semibase" is more logical than "subbase" (since the latter term seems to indicate that we indeed have some kind of base); but this is of no real importance, compared to usage. Thus, for the time being, I leave the term "semibase" on the wikibook page; but if this term was invented there and is only used there, then it should eventually be replaced.

On the other hand, if anyone has seen that "semibase" does appear in the literature (wikibooks uncounted:-), then the term should be mentioned at the start of this article, and a redir written from Semibase. Thus, my question: Has anyone seen this alternative name used somewhere else? JoergenB (talk) 18:05, 13 August 2015 (UTC)

Restoring an older version

I have restored a older version of the article for the following reasons.

The difference between the two versions consists essentially of approximately 50 edits by the same user, Mgkrupa. The edit summaries do not allow to know if there is a significant content change between the two versions. Most changes deal with the presentation of the content. Many are simple changes of styles that ar forbidden by MOS:VAR, and also make very difficult to read the source of the article. All together, the edits change the encyclopedic style into a textbook style (see WP:NOTTEXTBOOK). In particular, definitions, theorems and other results are systematically presented in a quoted form. In Wikipedia, this may be convenient for the most importent results that require to be emphasized. Using this, as it has been done here, for recalling standard notations and results that are described in other articles is very confusing, since it makes difficult to find the main results (there are few in this article) and the significance of the definitions (this should be the most important part of this article).

It is possible that some of these numerous edits are content improvement, but checking this is too time consuming, as this is not clear from edit summaries. D.Lazard (talk) 09:38, 4 October 2020 (UTC)

Since you reverted my edit that fixed several technical details and removed redundant information, please fix these issues that you have reintroduced.Mgkrupa 17:12, 5 October 2020 (UTC)

To editor Mgkrupa: It is to you to fix things that you think that need to be fixed. It is not the fault of other editors if you mix bold edits that are likely to be controversial with technical edits that hopely are not. Here there is a reorganization of the structure, and such reorganizations deserve always to be discuted. There is also, if I believe your previous post, several technical details, which are difficult to identify without spending time that would better be used elsewhere. So, if there are things to fix, fix them one after the other, with an edit summary that explains clearly each edit. If you think that this could be done better by another editor, explain clearly each issue in separated edit requests. D.Lazard (talk) 18:24, 5 October 2020 (UTC)

Thank you D.Lazard. Well said! PatrickR2 (talk) 04:35, 16 April 2022 (UTC)

Verbal proofs

The proof of the Alexander Subbase Lemma here is good and I don't want to change the standards of proofs in mathematics, yet it might (or might not) be convenient for readers of an encyclopedia to read proofs in a different style, and I'll just post this as an example for the Alexander Subbase Lemma.

Proof. Unless there are no topological covers of X without finite subcovers at all, Zorn's Lemma states that there are such covers that are maximal, i.e. do possess a finite subcover, when another open set is added to them.

When such a cover is restricted to its subbasic elements, it is no longer a cover, since otherwise it would possess a finite subcover under the conditions of the lemma. Hence there is at least one point of X that isn't covered by the subbasic elements of the cover and neither is some environment from the cover that covers that point, henceforth called the missing environment. However, the missing environment has subbasic generators containing the point in question. Hence these generators are not part of the cover and adding them to the cover provides us with a finite subcover in each case. Since they themselves are only finitely many, the union of the subsets of the cover needed to form these subcovers is still a finite set. But then it covers everything except an area within the intersection of the generators, which is itself contained in the missing environment. Hence adding the missing environment to the union provides us with a finite subcover of X, showing that under the conditions of the sublemma there are no topological covers of X without finite subcovers.

Alternate definition section

The "Alternate definition" section (added 11 Apr 2005 to the article) says that in some sources the definition of subbase requires that the collection of sets cover X. As a sample source (added 13 Jun 2013) it gives a reference to the book "Topological Methods in Chemistry", written by two chemists from industry.

1. I propose to remove that source for the following reason. The authors are not mathematicians. As they themselves say in the preface of the book:

"The purpose of this book is to introduce a new method and language for investigating molecular structure based on finite topological spaces. It is not intended as a mathematical text ... We have eschewed the ... style typical of mathematical works in favor of the type of mathematical discourse commonly used in physical sciences."

I don't want to distract anything from the value of the book for its intended purpose, but I don't think it can be taken as an authoritative source for standard terminology/definitions in mathematics.

2. Can anyone provide a good mathematics source for this alternate definition? The standard sources I have checked (Willard, Engelking, Dugundji, Arkhangelskii, ...) all use the first definition, i.e., without requiring that the collection of sets cover X. If there is no such source, maybe the alternate definition should be removed from the article? PatrickR2 (talk) 05:26, 16 April 2022 (UTC)

I removed the reference to "Topological Methods in Chemistry". PatrickR2 (talk) 21:12, 21 April 2022 (UTC)