Definition talk:Monotone Class

There are various compound objects out there consisting of various systems of sets, with various degrees of closure: closed under finite intersections, countable intersections, uncountable intersections, etc., and the same for unions. For example, sigma-algebras, topologies, measure spaces. (This definition is another.)

Wonder whether it's worth putting a page together summarising these by transcluding the relevant bits and indicating a hierarchy: such-and-such is contained in so-and-so, etc. (e.g. "a finite topology is a monotone class" and so on).

See Sequence of Implications of Separation Axioms, for example.

It would be an interesting project, and I've never seen such a thing attempted. But it stands to reason that someone out there has at least thought about it. --prime mover 14:57, 24 March 2012 (EDT)

I think this is a good idea; it will allow for a bunch of proofs to be bundled together like is eg. done for general relations, homomorphisms and so on. This will improve the structure and coherence of PW, which is a good thing. So, when do I have time? Wait, there's so much still waiting... --Lord_Farin 17:53, 24 March 2012 (EDT)

Time is an illusion caused by lack of acid. I thought everyone knew that. --prime mover 19:21, 24 March 2012 (EDT)

"On $X$" feels best intuitively so I've changed it thus. Plenty of links on the net but I haven't found one that uses any preposition in relation to $X$. I feel it's up to us to innovate. --prime mover 04:19, 25 March 2012 (EDT)

Is the way I presented the stuff at the moment suitable for handling multiple definitions, or should another template/style be invoked? --Lord_Farin 16:04, 22 May 2012 (EDT)

For uniformity of reference, a style using 'Definition 1' and so on could be developed, similar to what is done for proofs. I'm starting to support this approach. --Lord_Farin 16:08, 22 May 2012 (EDT)

Sounds like a good idea to me. It's a different situation, though - alternative definitions are rarer than alternative proofs, and we also have the complication of: sometimes the alternative definition is a different definition, that is, it defines a subtly different object. In which case we have to be careful.

I would actually dispute that the "also defined as" given here is actually a different definition as a countable indexing set is so "obviously" equivalent to $\N$ that it makes no difference. After all, the proof of equivalence is little more than "$\N$ is countably infinite, therefore the definitions are equivalent", am I right? --prime mover 16:34, 22 May 2012 (EDT)

That's one implication; the other is established by enlarging a union over $I$ to one over $\N$ by the given injection $I \to \N$ (here, it is critical that $I \ne \varnothing$ as is assumed). --Lord_Farin 17:01, 22 May 2012 (EDT)