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 $\mathcal P (X)$ (and $\mathcal P(X)$ 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)