Definition talk:Sub-Basis/Analytic Sub-Basis

Question
I see two possibilities here, and I'm not sure that it is clear which is the correct one:
 * $(1): \quad$ Steen and Seebach adopt the convention that $\bigcap \varnothing$ is the whole topological space. I'm not sure how this would fit in with Intersection of Empty Set, though.


 * $(2): \quad$ Steen and Seebach adopt the convention that a sub-basis covers the whole space. It appears that some authors do not require this. In this case, we should probably use the same convention for both Definition:Analytic Sub-Basis and Definition:Synthetic Sub-Basis.

--abcxyz (talk) 20:04, 30 September 2012 (UTC)


 * I don't understand your comment "I'm not sure how this would fit in with Intersection of Empty Set, though." That is exactly what Intersection of Empty Set says. $(1)$ holds.


 * However, I understand your confusion over the fact that if $\bigcap \varnothing = \mathbb U$ then an empty set covers the whole space - I believe that the definition of a "cover" should specifically exclude this possibility by insisting that an element of a cover should not be empty. --prime mover (talk) 06:16, 1 October 2012 (UTC)


 * I've added the (probably crucial!) missing words from S&S (my mistake back when I first posted it) - feel free to inspect it. --prime mover (talk) 06:16, 1 October 2012 (UTC)


 * Sorry that I didn't make it clear what I was referring to. The argument in Intersection of Empty Set (in fact) shows that $\forall x: x \in \bigcap \varnothing$, but then we are confronted with Russell's paradox if we assume that $\bigcap \varnothing$ is a set (such as $X$, for instance).


 * I don't understand your comment on covers. How can the empty set cover $X$?


 * Yes, the change looks good. I don't think we need to explicitly mention $\varnothing$, since $\bigcup \varnothing = \varnothing$ anyway. --abcxyz (talk) 05:01, 5 October 2012 (UTC)


 * Apologies, I completely misunderstood everything you said, from your first point onwards. Can you try again? --prime mover (talk) 05:29, 5 October 2012 (UTC)


 * I don't see how we can say that $\bigcap \varnothing = X$, since we have $\forall x: x \in \bigcap \varnothing$ by vacuous truth. (There is no restriction on $x$, as far as I know.) --abcxyz (talk) 16:58, 5 October 2012 (UTC)

For the purposes of topology, the universe of discourse is taken to be restricted to $X$. So every $\forall x$ is taken to read $\forall x \in X$. Does this resolve your problems? --Lord_Farin (talk) 17:04, 5 October 2012 (UTC)


 * I think I kind of know what you mean; so do we define $\bigcap \varnothing = \left\{{x \in X: \forall y \in \varnothing: x \in y}\right\}$? That is, is the intersection implicitly taken to be "carried out in $X$" (so to speak), even when there's no mention of $X$ in the notation $\bigcap \varnothing$?


 * Also, how can we take $X$ to be the universe while we concern ourselves with $\mathcal P \left({X}\right)$ and its subsets? --abcxyz (talk) 18:40, 5 October 2012 (UTC)


 * The subsets of $\mathcal P \left({X}\right)$ (and $\mathcal P \left({X}\right)$ itself) are just collections of subsets of $X$. These thus fit inside the definition of universe. It's not that ridiculous, and after all it's just a convention. We really do want to talk only about elements of $X$, therefore I think it's quite natural as well. --Lord_Farin (talk) 08:06, 6 October 2012 (UTC)


 * Maybe it should be made explicit on the definition page that we consider $X$ as the universe, and therefore take the convention $\bigcap \varnothing = X$? --abcxyz (talk) 18:45, 6 October 2012 (UTC)

I'm a fan of avoiding definitional nitpickery whenever that can be done conveniently and rigorously. The easy way to do so in this case is to avoid the case of empty intersections altogether. There are two ways to do this:
 * 1. Let $\tau'$ contain $X$ and the intersections of nonempty finite subsets of the proposed sub-basis. (This works if we don't require the sub-basis to cover $X$).
 * 2. Let $\tau'$ contain just the intersections of nonempty finite subsets of the proposed sub-basis. (This works if we require the sub-basis to cover $X$).

The advantage of (2) is that the underlying set does not need to be specified. I consider this insignificant (the underlying set is virtually always specified anyway). The advantage of (1) is that one does not need to ensure that a sub-basis contains the underlying set, which is in practice more troublesome.

Next: this language of "analytic sub-basis" vs. "synthetic sub-basis" seems to me a linguistically confusing way to describe the actual concepts involved. The most basic concept is actually topology generated by sub-basis (this is defined here as Definition:Generated Topology). The subordinate concept is subbase for topology (here called Definition:Analytic Sub-Basis). I have to think there's a way to present these concepts without such intimidating language and such repetition. --Dfeuer (talk) 20:41, 2 February 2013 (UTC)


 * As it is, it serves well enough. Please leave it alone. --prime mover (talk) 20:55, 2 February 2013 (UTC)


 * I understand that "leave it alone" is your primary philosophy, but I believe that if someone like myself who is already familiar with these concepts comes to the page and goes "WTF is all this confusing stuff about, and which thing do I use when?" then it needs changing. I also understand that Steen &amp; Seebach happened to be the first text you encountered about topology, but its mini-exposition topological concepts is not intended as an introduction and does not serve as a good one. That text is designed as a list of badly-behaved topological spaces with multiple tables indexing into them. It serves that function admirably. Don't push it past its limits by insisting that everything in general topology be defined as they do it. --Dfeuer (talk) 21:04, 2 February 2013 (UTC)


 * As suggested before, then please fill the gap and start covering Munkres proper. You're not getting anywhere by ranting savagely at many pages in an unstructured manner. It will make people disinclined to listen to you. Also, please mind your language (if you can't, please take a five minute walk before responding). --Lord_Farin (talk) 21:08, 2 February 2013 (UTC)


 * (Aah let the guy potty-mouth away, if it helps.)


 * Not only S&S but also Sutherland split it into two separate concepts. I repeat what I said earlier on a different thread: just because it does not make sense to you does not mean that it is of substandard comprehensibleness. The problem may reside with you. "... already familiar with these concepts ..." - you can't be or you wouldn't have a problem with this. --prime mover (talk) 21:10, 2 February 2013 (UTC)


 * Could we please return to a realm where a sensible and rational discussion is possible? --21:11, 2 February 2013 (UTC)