Date: Apr 14, 2012 12:33 AM
Subject: Re: Limit by limit of curves
>> Ok, I'm finished with this thread.
>> It's clear that we disagree.
>I don't even know what we are disagreeing about.
>You are insisting that for you to accept the notation there
>should be some uses of it in the books in the context of general topological spaces.
No -- that's not what I said.
What I said (many times already) was this:
If a notation is non-standard, you should _define_ it before
using it -- that's all.
I asked if the notation was used in some well regarded textbook
in the way you're using it only to judge whether the notation
could be regarded as standard or not. The lack of response
suggests that it's not standard, and if that's the case, one
might ask why all the many authors avoid your notation, opting
instead for versions defined using either
(1) sequences (for appropriate spaces):
f(x_n) -> q as x_n -> p
(2) nets (for general topological spaces):
f(x_lambda) -> q as x_lambda -> p
(3) filters (for general topological spaces):
f(F) -> q as F -> p
Why don't they simply define
f(x) -> q as x -> p
using neighborhoods? Why all the fuss with nets and filters?
Possibly because they've decided that the arrow notation
should mean that some actual entity (a sequence, net, or filter)
is approaching something.
In any case, as far as I'm aware, they don't use your notation,
but as I've said many times (now pay attention!), using a
non-standard notation is fine provided you define it before
I'll say it again -- using a nonstandard notation is OK if
you think there's some advantage, but using a non-standard
notation without defining it invites misinterpretation.
But do whatever you want -- I don't want to argue about this