Talk:Synthetic Basis formed from Synthetic Sub-Basis

I had great difficulty getting the pages relating to Definition:Generated Topology to fit together and flow properly. As such there is plenty room for improvement and I welcome any attempts to bring some shape to this. So my view is: have a go, see if you can make this section work. --prime mover (talk) 08:30, 23 September 2012 (UTC)


 * Why the change from A to F? Just curious, didn't seem to be consistent with general policy. Do we use F everywhere else in this area? --prime mover (talk) 16:31, 22 November 2012 (UTC)


 * I just didn't want to use the same symbol on Definition:Analytic Sub-Basis to define both $\mathcal B$ and $\tau'$. (By the way, $\mathcal F$ stands for "finite.") --abcxyz (talk) 16:36, 22 November 2012 (UTC)

I think there may be something wrong with this proof. A synthetic basis is defined to be a set of some sorts, but if the universe is an element of $\mathcal B$, then $\mathcal B$ can't be a set. Suppose that $\mathcal B$ is a set, then by the axiom of union, $\bigcup \mathcal B$ is also a set. The universe isn't a set; a contradiction soon follows.

One way to fix this is to change the definition of a synthetic sub-basis so that it is also a cover for $X$, then change the construction of $\mathcal B$ to
 * $\displaystyle \mathcal B = \left\{{\bigcap \mathcal F: \mathcal F \subseteq \mathcal S, \, \left \vert{\mathcal F}\right \vert \in \mathbb N_{\gt 0}}\right\}$

Here we have the condition that the cardinality of $\mathcal F$ is both finite and positive, so you won't be able to do anything weird like $\bigcap \varnothing$.

Please let me know if I'm wrong or anything. --HumblePi (talk) 16:33, 13 January 2017 (EST)


 * First of all, sorry for the long delay... But to the content. The implicit assumption here is that we are working within $X$ as our universe where $\bigcap$ is concerned, so that $\bigcap \varnothing = X$. I believe this addresses all your concerns.
 * We could argue whether this assumption needs to be explicated. I think it's pretty clear, but I'm also not opposed to doing so. However, changing the definition of subbasis would be overkill. &mdash; Lord_Farin (talk) 13:00, 24 April 2017 (EDT)


 * Thanks for the reply; I really don't mind if it takes a while.


 * I'm new to this idea of there being different universes, so I would just like to make sure I understand this. When you take a set $X$ to be the universe, that just means that your quantifiers ($\forall$ and $\exists$) range over $X$, and this is only done for the intersection $\bigcap \varnothing$. Correct? --HumblePi (talk) 18:09, 24 April 2017 (EDT)


 * Exactly. And here it is taken implied because $\mathcal S \subseteq \mathcal P(X)$. &mdash; Lord_Farin (talk) 11:35, 25 April 2017 (EDT)