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I noticed that presently topological basis redirects to basis in functional analysis instead of to the entry topological base. This seems dangerous. I’d like to change that redirect.
Sounds good to me
Okay, I have fixed it.
Can we at least have some disambiguation? The term ‘topological basis’ as defined at basis in functional analysis is standard, and one can always say ‘base’ in topology. (A base and a basis are really different things.)
Please do add a disambiguation sentence.
But is the use so standard as you say? Asking Google for “topological basis”, then it returns plenty of webpages that use it in the sense of “topological base”. The only page that I see Google give which uses it in the sense of “linear basis” is… our very own $n$Lab entry “basis in functional analysis”.
I say “basis elements” when referring to elements of what Toby calls a topological base. I first learned topology from the book by Munkres, and he writes “basis” for what Toby calls a base. I never knew there was supposed to be a rule about usage as given in Toby’s link.
It's true that the phrase ‘topological basis’ is used much more often to mean a topological base than to mean a topological basis, but this is just because people talk about the former much more often than they talk about the latter. It's still terrible terminology.
ETA: The linear-algebra analogue of a base is a spanning set, not a basis. A basis is an independent spanning set (a concept that has no analogue in topology).
It’s still terrible terminology.
But why? Who else thinks that?
Can you point to something in the literature that attests to the usage guideline mentioned in Warning 1.1 (or whatever it was) in your prior link?
I’ve been thinking a little more of ’base’ vs. ’basis’, and maybe I understand your preference to some extent. But only some.
I don’t know about a general theory of base-s, but there is at least one reasonable theory of basises (in this one instance I am grateful to Jim Dolan for that term), described at matroid. It must be said that this is a somewhat restricted theory, however. One of the guiding examples is of course the model theory of vector spaces, and this very likely was the reason for choosing ’basis’ to name one of the main concepts of the theory. Another guiding example is the model theory of algebraically closed fields, but there we find ’transcendence base’ and ’transcendence basis’ both in wide circulation (I think I’ve heard the former more often, but I’m just one person).
It seems to me that the term ’basis’ in algebraic situations is overwhelmingly used for linear or vectorial situations. I don’t think I’ve ever heard “basis for a free group” or “basis for a polynomial algebra” (the latter would be pretty confusing, since you hear rather more often of e.g. basis of monomials, which is again a linear situation). I would consider “orthonormal basis” for Hilbert space again as fitting inside the linear family (with this case having a little topology mixed in).
As for the examples listed under base: Google searches turn up “basis for a topology”, “filter basis”, “basis of a sigma-algebra”, and “basis for a uniformity”. Isbell also speaks of an entourage basis. I haven’t checked statistics for these word usages.
So my gut feeling is that there is little consensus on base vs. basis, except for the linear cases as mentioned. I’m not convinced that this is a basis (ha!) for extrapolating to proper rules of usage.
In Slavic languages we do not distinguish words base and basis (baza or alike), but by a topological basis we would mean one in functional analysis as opposed to an algebraic basis but we would say a base/basis of topology (notice the word order!) for the notion in topology and similarly for a local base we would say fundamental system of neighborhoods (of a point). When I came to US I talked pre-basis (predbaza) for what is literal translation in Slavic languages of notion standardly in English called a subbase of topology and (of course!) people did not understand what I was talking about. Then I learned to say base and not basis. Now 20 years later you are telling me to go back and talk basis like I did with my Slavic accent and not distinguish from the word in functional analysis ?
Zoran, are you addressing someone in particular in your last sentence?
In case it’s me, I’m not telling you to do anything. If you want to write “base of a topology” in the nLab, then please do – I’m sure everyone will understand perfectly. At the same time, I prefer for my own part to keep using the phrase “basis for a topology”, as that’s what I grew up with, and I don’t believe there is a consensus either way of what is “right”.
No, I was not addressing anybody, but trying to describe/express vividly as possible my experience/environment and bewilderment as a witness of the confusion in wider context (somebody above alluded to British vs. American, I went further beyond).
I have added a disambiguation line at the very top of topological basis, and a warning at the beginning of basis in functional analysis.
Todd, I can provide references for people using ‘base’ (but I probably don't need to), and we can presume that they have a reason for doing so (although admittedly it could always turn out to just be copying somebody who copied somebody who copied somebody who made a mistake). But I have no reference for my ‘terrible terminology’ claim in #7. I just have an argument for it, which is the edit added to #7. They're just not the same kind of thing, so it's best to use terminology that confuses them as little as possible.
Hi Toby. We might be more in agreement than appears. I agree there’s no need to give evidence of people using ’base’ these ways (there’s no shortage of that), and maybe more importantly I understand the conceptual distinction you wish to introduce. So I do see your reasons for your preferences, and I’m glad to learn your point of view.
We might disagree on whether it’s a sin for people to be saying ’topological basis’. I’m just not convinced that the usage for the linear cases should dictate the global usage across mathematics (which I don’t find to be universal across algebra either), and I also think there’s essentially zero chance that people who say ’topological basis’ are conflating anything in their own minds or sowing confusion in the minds of others. But at least I know that when I say ’basis for a topology’ (I have no current plans of changing that) in the nLab, I’ll know to link back to base and not basis, and I hope that’s good enough. :-)
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