Talk:Subbase
From Wikipedia, the free encyclopedia
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.
--<<unsigned, undated>>--
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
[edit] 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
[edit] 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)
