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started a disambiguation page basis
Disambiguated the disambiguation page basis :)
Thanks.
The disambiguation of “base” is something I am thinking actually the title of that entry might want to reflect.
The disambiguation of “basis of a topology” is not so much one, is it? A basis of the Grothendieck topology of a category of open subsets of some topological space should correspond to a basis for the topology of that space.
Replaced “basis of a vector space” with “basis of a free module”.
I wrote up a page basis of a free module.
Cleaned up the formatting a little.
It’s a minor point, but if characterising a basis of a free module as an isomorphism between it and a “standard” free module, then I would write the isomorphism in the other direction; namely, as $R[B] \to M$. I much prefer mapping out of free things than mapping in to them, even if they are isomorphisms!
Yes, I think the titles of the pages base and basis for a topology are pretty confusing. I would maybe call them “base for a topology” and “basis for a Grothendieck topology” or something.
Actually, is “basis for a (Grothendieck) topology” an unintentional duplication of Grothendieck pretopology that should be merged?
Mike,
I see, you are right. So I created quite some mess. Sorry. Will try to clean it up later.
My wife just looked over my shoulder and wants to know why we’re not using the word “clarification”, seeing as that’s actually an English word (unlike “disambiguation”).
“Disambiguation” is so an English word. It is also more specific than “clarification;” one can clarify things in many ways, but one disambiguates them particularly by removing a source of ambiguity.
Andrew’s wife is probably referring to the famous essay by Orwell, “Politics and the English Language”.
I like that essay quite a lot, but why is Andrew’s wife “probably referring” to it?
@Urs #3
actually a basis for a topology is only a coverage of the category $\mathcal{O}$ of open sets of a space. A basis for a Grothendieck topology (a pretopology) on $\mathcal{O}$ is the usual thing: a cover of an object is just an open cover of the open set.
@Todd: I dunno, but “disambiguation” is one of those Latin/Greek monstrosities that Orwell talks about in the essay. I didn’t give any thought to whether or not “clarification” is similar in this respect, and I don’t know its etymology offhand, but if it is not a loan-word, then my comment might make sense.
Oh, I see. Yes, I believe “disambiguation” is a Greek-Latin hybrid (whereas “clarification” is not), so okay. It’s by no means clear that Andrew’s wife was thinking of the essay (or has even read it), but at least I see what you were driving at.
I feel bad that a presumably offhand comment by Andrew’s wife is being subjected to such scrutiny, but FWIW I note that the earliest citation of “disambiguation” in the OED seems to be from 1827 (unless my worsening eyesight deceives me).
Edit: I’m wrong! “ambi-” is Latinate, and it was silly of me to think otherwise. So scratch that.
By “Greek/Latin”, I meant “derived from either greek or latin”, mainly transmitted through French loan words that came into the language through the Normans. Anyway, I was wrong, because “clarification” also comes from latin.
A basis of the Grothendieck topology of a category of open subsets of some topological space should correspond to a basis for the topology of that space.
Really ? Not a prebasis ? I mean pullbacks are intersections. In prebasis you can use intersections as well; in basis of topology you can not. There is also a basis of neighborhood of a point (or fundamental system of neighborhoods or basis of local topology).
@Zoran - see my #13. I agree that there is also the concept of a basis of neighbourhoods - it is my favourite approach to defining a topology! But I don’t know how to reconcile the Grothendieck topology approach and neighbourhood bases. The latter after all uses points of the space, and this is not a good notion except for sober spaces, because for a non-sober space the open neighbourhoods aren’t enough to find the point again.
And see my #17.
So to straighten this out once and for all, I now created an entry basis for a topology after all, which discusses the topological notion and highlights its relation to the topos-theoretic notion.
I do not understand 19 and 20. I objected to base vs. prebasis/subbase (both notions from general topology) and you point me to the diuscussion on the comparison with Grothendieck topology. This has nothing to do with my objection. Namely if you generate by pullbacks/intersections that means not only unions then it is subbase in classical case, not base.
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