From Newsgroup: sci.physics

Peter Fairbrother <peter@tsto.co.uk> wrote or quoted:

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?

That might not be the kind of statement you could really prove,
since you would need some shared sense of what continuity means
"before it has a definition". In math though, a word only gets
its meaning once it is pinned down by a definition.

Continuity gets defined for topological spaces, and that is
a very broad definition, since lots of the spaces people care
about in practice are also topological spaces.

Nobody's managed to water down the conditions any further and
still come up with a definition of continuity that lines up
with what mathematicians expect.

2] what other structures on spaces (considered as sets of points) give >notions of continuity?

Well, I actually know a structure that is not a topological space,
but it still has its own idea of "continuity". The elements
are not really thought of as "points" anymore. It comes from
lattice theory and its connection to denotational semantics.

To really get that definition, you need a solid book on the topic,
but it goes something like: "If D and E are cpo's, the function
f is continuous iff it is monotone and preserves lubs of chains,
i.e., for all chains d0 c d1 c . . . in D, it is the case that
f(U(n>=0): d_n) = U(n>=0): f(d_n) in E.".

The term was probably picked because of the loose analogy to
continuity in topological spaces, and there was no real risk
of mixing the two up.


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From Newsgroup: sci.physics

On 24/08/2025 10:10, Stefan Ram wrote:

Peter Fairbrother <peter@tsto.co.uk> wrote or quoted:

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?

That might not be the kind of statement you could really prove,
since you would need some shared sense of what continuity means
"before it has a definition". In math though, a word only gets
its meaning once it is pinned down by a definition.

Continuity gets defined for topological spaces, and that is
a very broad definition, since lots of the spaces people care
about in practice are also topological spaces.

Nobody's managed to water down the conditions any further and
still come up with a definition of continuity that lines up
with what mathematicians expect.

So, simplest known structure, rather than simplest possible structure.
Thanks, that's about what I expected.

2] what other structures on spaces (considered as sets of points) give
notions of continuity?

Well, I actually know a structure that is not a topological space,
but it still has its own idea of "continuity". The elements
are not really thought of as "points" anymore. It comes from
lattice theory and its connection to denotational semantics.

To really get that definition, you need a solid book on the topic,
but it goes something like: "If D and E are cpo's, the function
f is continuous iff it is monotone and preserves lubs of chains,
i.e., for all chains d0 c d1 c . . . in D, it is the case that
f(U(n>=0): d_n) = U(n>=0): f(d_n) in E.".

The term was probably picked because of the loose analogy to
continuity in topological spaces, and there was no real risk
of mixing the two up.

Yikes! I know a bit about lattices as groups (I am a cryptologist), but
not as abstract algebra. Though I can read that, but not grok it. I will
maybe get a book ..

Ta again

Peter Fairbrother

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From Newsgroup: sci.physics

Peter Fairbrother <peter@tsto.co.uk> wrote or quoted:

Yikes! I know a bit about lattices as groups (I am a cryptologist), but
not as abstract algebra. Though I can read that, but not grok it. I will >maybe get a book ..

In the meantime, it came to my mind, there might be a generalization
of the concept in category theory. So, I asked the chatbot and got:

|In classical topology, continuity of a function between
|topological spaces means the preimage of every open set is open.
|
|In lattice theory and denotational semantics, continuity
|often refers to preserving directed suprema or certain
|order-theoretic limits.
|
|Category theory abstracts these ideas by defining a /functor/
|to be /continuous/ if it preserves limits. More precisely:
|
| A functor F: C --> D is called /continuous/ if it preserves all
| small limits that exist in C. That means for any diagram D: J --> C
| whose limit exists in C, the image under F of this limit
| is isomorphic to the limit of the composed diagram F o D in D.
|
| This notion extends to various categories including posets,
| where a poset is continuous if it is a continuous category in
| the sense of having particular filtered colimits and adjoints
| related to ind-objects. The classical notion of continuous
| posets fits as a special case of continuous categories.
|
| Thus, continuity in category theory generalizes both continuity
| in topological spaces and continuity of functions on lattices (as
| used in denotational semantics and domain theory). It unifies these
| via the preservation of universal constructions like limits.
|
| Continuous categories and continuous functors form a rich
| theory that includes locally finitely presentable categories,
| Grothendieck toposes, and higher-categorical analogs, showing
| deep connections across topology, order theory, and semantics.
|
|Hence, continuity is abstracted by category theory as the
|preservation of limits by functors, which generalizes the
|classical and order-theoretic notions of continuity.
.

"Continuity in category theory", with or without quotation
marks, can be given to web search engines to learn more . . .


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From Newsgroup: sci.physics

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?

2] what other structures on spaces (considered as sets of points) give notions of continuity?

Centuries ago in real analysis I ran across the theorem that:

"A continuous function is one whose inverse maps open sets to open sets"

which is a special case, I think, of the Heine-Borel theorem, which in
its more general form may be relevant to your question.

William Hyde

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From Newsgroup: sci.physics

William Hyde <wthyde1953@gmail.com> wrote or quoted:

"A continuous function is one whose inverse maps open sets to open sets"

It might be better to ask that the preimages of open sets come
out open, because if you think about the sine function, you
can still use that test on it even though it does not have an
inverse, since that wouldn't be single-valued.

(I'm not too sure about this, but that's just what came to mind.)

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From Newsgroup: sci.physics

Stefan Ram wrote:

William Hyde <wthyde1953@gmail.com> wrote or quoted:

"A continuous function is one whose inverse maps open sets to open sets"

It might be better to ask that the preimages of open sets come
out open, because if you think about the sine function, you
can still use that test on it even though it does not have an
inverse, since that wouldn't be single-valued.

(I'm not too sure about this, but that's just what came to mind.)

We're getting well beyond the limits of my memory here. All I recall of general topology is that I tried to use Hausdorf axioms to prove just
about anything.

As for measure theory, I have nightmares of hour long classes which in
the end succeeded only in proving that

"The unit sphere is closed in the weak* topology".

and I can no longer recall why that was of any importance.

William Hyde
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From Newsgroup: sci.physics

William Hyde <wthyde1953@gmail.com> wrote or quoted:

As for measure theory, I have nightmares of hour long classes which in
the end succeeded only in proving that
"The unit sphere is closed in the weak* topology".
and I can no longer recall why that was of any importance.

Yeah, I think one of the best initiation rites is when
they throw this at you: "Show that open balls are open!".


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From Newsgroup: sci.physics

William Hyde <wthyde1953@gmail.com> wrote or quoted:

Stefan Ram wrote:

William Hyde <wthyde1953@gmail.com> wrote or quoted:

"A continuous function is one whose inverse maps open sets to open sets" >>It might be better to ask that the preimages of open sets come

out open, because if you think about the sine function, you
can still use that test on it even though it does not have an
inverse, since that wouldn't be single-valued.

We're getting well beyond the limits of my memory here.

Oh, that's not really a big deal. The whole point is just that
every function in math, whether it's topology or whatever else,
has to be set up so each x only matches with one y.

Figure 1 shows that with the sine function.

| .+++.
| .+ ^.
y |<---.+ ^.
| +| +
| | | +
| .' | .
| | | +
| .' | -
| + | |
|. | -
|+ | |
|' | -
+++++|++++++++++++++++++
x

But the other way around, a lot of functions end up giving you a
bunch of possible values, so you no longer have a proper "inverse
function".

Figure 2 shows that again with the sine function.

| .+++.
| .+ ^.
y |----.+-----^.
| +| |+
| | | | +
| .' | | .
| | | | +
| .' | | -
| + | | |
|. | | -
|+ | | |
|' V V -
+++++|++++++++++++++++++
x_0 x_1

You can still talk about points like "x_0" and "x_1" in the
figure, but they're just "preimages", not actual outputs of
an "inverse function".


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From Newsgroup: sci.physics

On 26/08/2025 11:00, Stefan Ram wrote:

William Hyde <wthyde1953@gmail.com> wrote or quoted:

Stefan Ram wrote:

William Hyde <wthyde1953@gmail.com> wrote or quoted:

"A continuous function is one whose inverse maps open sets to open sets" >>> It might be better to ask that the preimages of open sets come

out open, because if you think about the sine function, you
can still use that test on it even though it does not have an
inverse, since that wouldn't be single-valued.

We're getting well beyond the limits of my memory here.

Oh, that's not really a big deal. The whole point is just that
every function in math, whether it's topology or whatever else,
has to be set up so each x only matches with one y.

Figure 1 shows that with the sine function.

| .+++.
| .+ ^.
y |<---.+ ^.
| +| +
| | | +
| .' | .
| | | +
| .' | -
| + | |
|. | -
|+ | |
|' | -
+++++|++++++++++++++++++
x

But the other way around, a lot of functions end up giving you a
bunch of possible values, so you no longer have a proper "inverse
function".

Figure 2 shows that again with the sine function.

| .+++.
| .+ ^.
y |----.+-----^.
| +| |+
| | | | +
| .' | | .
| | | | +
| .' | | -
| + | | |
|. | | -
|+ | | |
|' V V -
+++++|++++++++++++++++++
x_0 x_1

You can still talk about points like "x_0" and "x_1" in the
figure, but they're just "preimages", not actual outputs of
an "inverse function".

Hmmm, as we are talking about sets here, rather than individual points
or values, does it matter whether a point in the target has more than
one inverse?

Peter F

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From Newsgroup: sci.physics

Peter Fairbrother <peter@tsto.co.uk> wrote or quoted:

Hmmm, as we are talking about sets here, rather than individual points
or values, does it matter whether a point in the target has more than
one inverse?

It really does not make much difference. The proper term would be
"relation" instead of "function". A relation counts as "functional"
if every x matches up with at most one value y. And "function" is
basically just the usual shorthand for saying "functional relation".

A function "f", so that "f(1)=3" and "f(2)=3":

f
x y
1 3
2 3

. The inverse "r" has the value columns swapped:

r
x y
3 1
3 2

This "r" is a relation, but not a function.

Wikipedia: "The inverse of 'f' exists if and only if 'f'
is bijective". The "f" from above is not bijective (because
it's not injective), so the inverse of "f" does not exist.

One might define a function "g" using sets, "g(3)={1,2}":

g
x y
3 {1,2}

. But this is not the usual meaning of "inverse function", because
"f(g(x))" would be undefined instead of "x". However, "{1,2}" is the
preimage of "3". Conveniently, the "preimage" /is/ defined as a set:

|preimage - Wiktionary:
|
|preimage (plural preimages). (mathematics) For a given
|function, the /set/ of all elements of the domain that
|are mapped into a given subset of the codomain; ...

. So, the preimage might exist even if no inverse exists.

I only went over the standard terminology here, the kind you
pretty much get in school. Of course, anyone can come up
with their own version if they want. Some authors talk about
"multivalued functions", but that usually ends up being read
as "set-valued functions". The "full inverse" of a function
then gets defined as one of those "multivalued functions".

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From Newsgroup: sci.physics

ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:

. The inverse "r" has the value columns swapped:

r
x y
3 1
3 2

This "r" is a relation, but not a function.

Wikipedia: "The inverse of 'f' exists if and only if 'f'
is bijective". The "f" from above is not bijective (because
it's not injective), so the inverse of "f" does not exist.

Above, I'm using "the inverse" and then later write,
"the inverse of 'f' does not exist.", which is a bit
inconsistent. So, it should be:

| . The inversion "r" has the value columns swapped:
|
|r
|x y
|3 1
|3 2
|
| This "r" is a relation, but not a function.
|
| Wikipedia: "The inverse of 'f' exists if and only if 'f'
| is bijective". The "f" from above is not bijective (because
| it's not injective), so the inverse of "f" does not exist.


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From Newsgroup: sci.physics

On Sat, 23 Aug 2025 09:47:48 -0700, The Starmaker
<starmaker@ix.netcom.com> wrote:

Are there canals on Mars or is that just...
a mistake of the word "canals" in a foreign country?

Maybe it means... Leave The Gun, Take The Cannoli.

They are looking for cannolies on Mars.

According to
https://en.wikipedia.org/wiki/Martian_canals

During the late 19th and early 20th centuries, it was erroneously
believed that there were "canals" on the planet Mars.

They were first described by a guinea wop astronomer Giovanni
Schiaparelli during the opposition of 1877, and attested to by later
observers. Schiaparelli called these canali ("channels"), which was mis-translated into English as "canals".

The guinea word canale (plural canali) can mean "canal", "channel",
"duct" or "gully".[1]

Leave The Gun, Take The Cannoli.


Now, How many YEARS did scientist believed there were canals on
Mars????

https://www.gettyimages.com/photos/martian-canals

Photographic / Photo Evidence of Lines (Canals) on Mars https://cdn.carleton.edu/uploads/sites/513/2020/08/583353_orig-scaled.jpg?resize=793,1024


Leave The Gun, Take The Cannoli.


So, How is Mars doing today? Anybody find any ancient Pyrmiads?

a coke bottle on the ocean beach...


If a pebble falls from outer space and makes a hole on somebodys
roof....NASA wants a billion dollars to save the world!

i never seen soooo much garbage from the Science community

A Meteorite Tore Through a Georgia HomeAs Roof. It Turns Out the Space
Rock Is Older Than Our Planet
A planetary geologist finds that the meteorite, which fell in June,
came from the main asteroid belt between Mars and Jupiter https://www.smithsonianmag.com/smart-news/a-meteorite-tore-through-a-georgia-homes-roof-it-turns-out-the-space-rock-is-older-than-our-planet-180987137/




Older than earth, came from between Mars and Jupiter

retards.








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From Newsgroup: sci.physics

On Tue, 26 Aug 2025 11:04:56 -0700, The Starmaker
<starmaker@ix.netcom.com> wrote:

On Sat, 23 Aug 2025 09:47:48 -0700, The Starmaker
<starmaker@ix.netcom.com> wrote:

Are there canals on Mars or is that just...
a mistake of the word "canals" in a foreign country?

Maybe it means... Leave The Gun, Take The Cannoli.

They are looking for cannolies on Mars.

According to
https://en.wikipedia.org/wiki/Martian_canals

During the late 19th and early 20th centuries, it was erroneously
believed that there were "canals" on the planet Mars.

They were first described by a guinea wop astronomer Giovanni
Schiaparelli during the opposition of 1877, and attested to by later >observers. Schiaparelli called these canali ("channels"), which was >mis-translated into English as "canals".

The guinea word canale (plural canali) can mean "canal", "channel",
"duct" or "gully".[1]

Leave The Gun, Take The Cannoli.

Now, How many YEARS did scientist believed there were canals on
Mars????

https://www.gettyimages.com/photos/martian-canals

Photographic / Photo Evidence of Lines (Canals) on Mars >https://cdn.carleton.edu/uploads/sites/513/2020/08/583353_orig-scaled.jpg?resize=793,1024

Leave The Gun, Take The Cannoli.

So, How is Mars doing today? Anybody find any ancient Pyrmiads?

a coke bottle on the ocean beach...

If a pebble falls from outer space and makes a hole on somebodys
roof....NASA wants a billion dollars to save the world!

i never seen soooo much garbage from the Science community

A Meteorite Tore Through a Georgia HomeAs Roof. It Turns Out the Space
Rock Is Older Than Our Planet
A planetary geologist finds that the meteorite, which fell in June,
came from the main asteroid belt between Mars and Jupiter >https://www.smithsonianmag.com/smart-news/a-meteorite-tore-through-a-georgia-homes-roof-it-turns-out-the-space-rock-is-older-than-our-planet-180987137/

Older than earth, came from between Mars and Jupiter

retards.

Why is it these guinea scientist seems to be all members of the
Mafia???

Guinea scientists don't say the big bang...they say bada bing, bada
boom!!!!

What came first, the bing or the boom?




Leave The Gun, Take The Cannoli.
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