Talk:Equivalence of Definitions of Topology Generated by Synthetic Sub-Basis

Advance warning: I'm about to restructure this so that the definition of a G.T. based on the sub-basis definition the main definition, and the 2nd definition an "also defined as", by renaming this result to something with iff in its title appropriately - but not tonight, Josephine, I'm too tired.

This ultimately agrees with both the definitions I have in the source works in front of me. --prime mover (talk) 22:09, 8 November 2012 (UTC)


 * What's wrong with the current "Def 1", "Def 2" structure (besides needing subpages)? We can't maintain the "ADA" section when more and more definitions of the subject crop up, ultimately necessitating another refactoring, mostly back to the way it is now. I don't get it. --Lord_Farin (talk) 22:45, 8 November 2012 (UTC)


 * In this particular case, because Def. 1 presents a practical method for creating such a topology (i.e. a: form all finite intersections, b: form all unions of those) whereas Def. 2 just gives a set of properties by which such a general topology may be recognised. And as such, Def. 2 just comes across as being less useful than Def. 1


 * The general case is that any set of properties interderivable with those given in the definition can be used as that definition instead of the original set. Hence it can be suggested that all of those could / (and some may argue "should") be added to the definition page as an "alternative definition". Taking this to its logical conclusion, we would have a definition page with multiple transcluded pages.


 * Actually, I can think of no logical argument against it, as long as it is structured appropriate. Def. 1, Def. 2, Def. 3 ... Def. 16 - as long as they are all transcluded as subpages (in the same paradigm as our Proof 1, Proof 2, ... then it might be argued that this site then provides the "added value" of gathering into one place all the "possible" definitions for a concept.


 * The only remaining argument against this is the current way we have of proving equivalence - at the moment this is done by a page with the lacklustre title "Equivalence of Definitions of ...", which may work when there are only two definitions, but when there are three or more, such a page degenerates into the potential anarchy which is (one I can't remember how to find now). We could structure it as: Definition 1, Definition 2 (including subsection "Justification" or some such, containing a link to a proof that this definition is equivalent), Definition 3 (including subsection "Justification"), etc. Although some of the "Justification" proofs are interlinked: 1 --> 2 --> 3 --> 1 etc., as long as the details of the complete flow of logic is given on each, to make them self-contained, that should work.


 * So I've persuaded myself to come towards agreement with you - so does it make sense to evolve the site into that direction? I may make a start on it with this page at the weekend approaching. --prime mover (talk) 06:47, 9 November 2012 (UTC)


 * ... or now, even. --prime mover (talk) 07:05, 9 November 2012 (UTC)


 * I'd say yes. A condition I'd like to impose is that for any definition added, at least one of the following holds:


 * It is defined that way in a source (unavoidably these definitions should be covered IMO).
 * It is particularly useful or insightful


 * It is often the case that there are multiple definitions of the same concept. It happens very often that it's easy to derive something is an instance of one, while it is very powerful (in a loose, sloppy sense) to know that it's an instance of the other. I'd have to think about whether we desire a lot of subtly differing definitions to clutter the page. I'll have to deliberate some more on the precise direction I want this to go, and how to implement that. Some ideas cropping up include, but aren't limited to: A page like Eq of Def, structured like:

Let $A$ be in the collection where things have a chance to meet the definition (e.g. a subset of $\mathcal P(X)$ for a topology). Then the following are equivalent: blablabla


 * thus explicating all definitions, then proving them equivalent in one stretch. I understand why you don't like the title, but this could be solved by inventing a better title rather than disposing of the page. The example you sought is undoubtedly Equivalence of Definitions of Exponential Function. It just occurred to me that we can maybe save ourselves by structuring along the lines of "Sequence of Implications of ..." (which, btw, doesn't seem very much less lacklustre than Eq of Def).


 * Returning to the case at hand: I'd say Def 2 has merit, in that it provides a direct justification of the nomenclature, similar to Definition:Sigma-Algebra Generated by Collection of Subsets (which in fact can be obtained directly by using transfinite induction up to the first uncountable ordinal; but that's just messy).


 * In conclusion, the main problem left to us is to provide a method to decide whether something should be merely a "characterizing property", so to speak, with simply its own page "X is Blabla iff Property", or that it be a full-fledged definition besides the one already given. In this case, I'd be happy to refer to the (unfortunately somewhat arbitrary) two points above; I feel they should be the main guidelines in this permanent dilemma. Maybe we could structure the EoD page as follows:


 * State eq of defs
 * Moreover, the following conditions characterize a (whatever's being defined):
 * Other characterizations, not worthy of mentioning on the def page


 * This scheme has the advantage of collecting together all means of proving a structure satisfies a definition. It's of course subject to change (e.g. the other chars could also be mentioned in some other way).


 * Finally, I suggest that we not pursue this scheme too eagerly; there's still plenty of other work to do without another stack of refactoring being thrown at us. Tagging pages with "refactor" should do for now. Maybe move this discussion to Main talk to inform the rest of the community? --Lord_Farin (talk) 09:37, 9 November 2012 (UTC)