From Newsgroup: sci.math
On 08/30/2025 10:16 AM, Ross Finlayson wrote:
On 08/30/2025 03:51 AM, Ross Finlayson wrote:
On 08/28/2025 12:50 PM, Python wrote:
Le 28/08/2025 |a 19:11, Mike Terry a |-crit :
On 23/08/2025 21:15, Peter Fairbrother wrote:
It is claimed that a topology O on a space M is the simplest
structure which affords a notion of continuity. Two questions:
1] is there a proof of this?
I doubt it. (It's not really a mathematical statement, more the sort
of motivational remark an author might make when introducing a student >>>> to the subject.)
2] what other structures on spaces (considered as sets of points)
give notions of continuity?
Well, there are metric spaces and uniform spaces. Both of those are
extensively studied, and give rise in a natural way to an underlying
topological space. (That's not to suggest there aren't other ways of
approaching continuity - those examples were just what came to mind.)
Mike.
I would say, nevertheless, that the topological definition is simpler
that the epsilon-delta one, as the first one is more general (a metric
implies a topology not the other way around). Another definition, not
taught as much, is the definition in term of filters (as in Bourbaki's
books) which can be seen as even more general as the topological
definition.
Sounds like somebody who says Dedekind cuts are field-reals, when
everybody knows that equivalence classes of sequences that are
Cauchy, are field-reals, and that if there are Dedekind cuts of
those, it's as of those, since the rationals are not gapless.
It's like somebody who says the initial ordinal assignment is the
cardinal, when the cardinal is the equivalence class of all those.
It's like somebody who doesn't know that continuous functions
are continuous functions whether or not they're over continuous
domains.
It sort of reminds of, "differential geometry". "To define our
immersions and submersions, so that a bunch of numerical methods
of Coates and Gregory apply, define 'function' as like a classical
function yet, not only having no asymptote in our coordinate
setting, also its tangent vector normal having no asymptote,
essentially eliminating values of infinity already, zero also".
And it's like, "maybe so, if every time you write 'f' and 'g'
for functions they're written 'f_DG' and 'g_DG' for this
subfield of 'differential geometry's'".
The widespread use of "almost all" or "almost everywhere" is
another sort of example, sometimes innocuous, others erroneous.
It's like 'what's convergence' and "well it results series
that are Cauchy" and "how exactly is that...", "any number of
convergence tests that aren't wrong". Ah, .... That though
is more about that there are inductive results that aren't
necessarily true "in the limit" as when things complete in
the limit.
The idea of a topology, that's also its own initial and final
topology, as with regards to spaces and points therein, and
whatever divides or adds those, sort of makes topology speak
to continuity instead of vice versa.
It sort of makes at least enough sense when "well it's sort
of interchangeable with the concept of points 'in' a line and
points 'on' a line, those being two different things".
It's like, in higher mathematics, when trying to discover or
having introduced notions of continuity and discreteness,
it's a great challenge since on a one hand there's the
Pythagorean, and at least almost all numbers are rational,
and on another or the Cantorian, at least almost all numbers
are transcendental, and that's a great divide, and by great
divide I mean it's impossible, without more widely and broadly
and by that I mean more comprehensively and inclusively,
figuring out how those two opposite results don't kill each other.
Then, when it gets worse and "well of course here's the modern
line of modern mathematics then here's some watered-down
intermediate, detached developments that then are to be
attached variously somewhere between the Pythagorean and
Cantorian, they don't really connect both ways since
they're detached from geometry or the arithmetic definitions,
now either way use these", and it's like, it's _neither_ way.
It sort of reminds of the quasi-modal in the logic or the
"material implication", say, which is good for mostly nothing
though as a bit-wise sort of operation it's sort of, you know,
parallelizable, the double-negative though or "not-material
not-implication", is _neither_, and people who bought or shilled
that are fools or frauds.
So, actually addressing the issues of course may arrive at
something like what I've developed here, where there is a
countable continuous domain after something like line-drawing,
and yes it's build-able in the standard and yes it's shewn
non-contradictory with the modern, thusly that otherwise
may be shewn Pythagoras and Cantor killing each other,
and leaving a great bloody mess.
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