Re: Seven deadly sins of set theory

On 02/13/2024 11:05 AM, Ross Finlayson wrote:

On 02/12/2024 07:32 PM, Ross Finlayson wrote:

On 02/11/2024 02:59 PM, Jim Burns wrote:

On 2/11/2024 9:17 AM, Ross Finlayson wrote:

On 02/10/2024 02:50 PM, Jim Burns wrote:

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and topologies of various sorts,
including at least one where
open and closed _are_ binary,

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A topological space X which doesn't have
at least two clopen (closed.and.open) sets,
X and ∅
isn't a topological space.
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A topological space ⟨X,𝒪⟩ is
a set X and a collection 𝒪 ⊆ 𝒫(X) of its subsets,
designated the set of _open_ sets of the space.
𝒪 is the topology of ⟨X,𝒪⟩
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The topological axioms for ⟨X,𝒪⟩ are:
1. The union of an arbitrary open.set.set 𝒜 ⊆ 𝒪
is the open.set ⋃𝒜 ∈ 𝒪.
2. The intersection of two open.sets B,C ∈ 𝒪
is the open set B∩C ∈ 𝒪.
3. ∅ and X are open sets.
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1. 𝒜 ⊆ 𝒪  ⇒  ⋃𝒜 ∈ 𝒪
2. B,C ∈ 𝒪  ⇒  B∩C ∈ 𝒪
3. ∅ ∈ 𝒪  ∧  X ∈ 𝒪
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The complement of an open set is a closed set,
and vice versa.
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The complement of open ∅ ∈ 𝒪 is open X.
∅ is clopen.
The complement of open X ∈ O is open 0
X is clopen.
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Still leaning toward continuous domains.
(From RF and JB about connected domains.)
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Now, "the standard open topology", is standard,
and it's open, and there are others.
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https://en.wikipedia.org/wiki/Topological_space#Definitions
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Here the neighborhood definition sort of sits well
with that these iota-values in the line-reals have
nearest-neighbors, as do those transitively ad infinitum,
so the notion of neighborhood is very natural.
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Those being open, their complements being closed,
those being open, their complements being closed, ....
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"A net is a generalisation of the concept of sequence.
A topology is completely determined if for every net
in X the set of its accumulation points is specified. "
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Basically any two points have a unique neighborhood
of them and the points between them, and, also all
points are in the same neighborhood.
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So, this way the topology of line-reals is about
the most simplest thing.
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There are others, ....
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The weakness of topology the property seems to be
that any two elements aren't neighbors, them not
being accumulation points and so on, there's something
to be said stronger than "connected" about that any
point is in a neighborhood with any other point that
isn't the entire set, and furthermore, that they are
in as many neighborhoods together as there are
supersets of their smallest neighborhood in the set.
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"When every open set of a topology Tau_1 is also
open for a topology Tau_2, one says that Tau_2 is
finer than Tau_1, and Tau_1 is coarser than Tau_2."
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So, here it's sort of that "the continuous domain
has the finest topology of the connected domains",
or that "an uber-fine topology has no finer topology".
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https://en.wikipedia.org/wiki/Compact_space
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"If a space X is compact and Hausdorff, then no finer
topology on X is compact and no coarser topology
on X is Hausdorff."
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https://en.wikipedia.org/wiki/Compact_space#Examples
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Now for these line-reals or ran(EF), ran(f), or iota-values,
the elements are identified starting from f(0), then as
with regards to there not being an f(\infty) there's as
above that the natural neighbors are from either [0, ...]
or to [..., 1].  So, there's a bit of a symmetry argument
about 1-EF that builds the same thing.
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"The Cantor set is compact. In fact, every compact
metric space is a continuous image of the Cantor set."
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What I have in mind here is that the
continuous domain has sorts of topologies,
that make it so, that every open set is
also a continuous domain, and, to make
it so, that only two open sets that are
"contiguous", are in this sort of topology.
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It's the initial and final topology.
There's no coarser or finer.
The indiscrete topology doesn't exist.
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This way then whenever any two sets with
this sort of topology intersect, then, they
form this same sort "continuous topology".
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It's not much to ask when most usual things
continuous and gapless and smooth and connected
have it this way, and it's also for making for about
the Euclid's smooth and Poincare's rough, reticulum,
that then those result while yet topologies, thusly
"degenerate continuous topologies", helping make
it quite clear and common what goes on in the
quasi-invariant and the discontinuous and the
almost-everywhere vis-a-vis the everywhere, and
these sorts things, to make it so that the definition
of topology here has this "continuous topology"
which maintains its usual properties as a topology,
but only holds its definition as a "continuous topology"
as for these "continuous domains", then that their
quite totally usual composition, is only what
results maintains this "continuous topology".
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I think it's very great and can be a real big deal.
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Of course this concept is buried deep in a
thread that's mostly fools talking about nothing,
and ex falso nihilum, but I usually just put
substantive material in these kinds threads,
then that establishing and maintaining it
a linear narrative and bulwark of definition,
just sort of results it's in the feed.
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So, here the notion "continuous domain" is
associated with a notion of a "continuous topology",
for the "continuum limit" of functions such as
make for modeling the line-reals, the field-reals,
and the signal-reals, these three sorts continuous
domains, helping to rehabilitate mathematics
formally, which has these same things as
continuum limits and ultrafilters and such,
that eventually contradict each other.
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So, let's consider these properties of a subset
of topologies which have properties as of
"continuous topologies" that relate the
connectedness of open sets that they're
always contiguous as so connected, thusly
as where the compositions of the elements
so connected, have the operation is natural,
that composition and decomposition maintains
the property of the continuous topology of
a continuous domain, up and down to the degenerate.
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(If you happen to have studied some category
theory, [0,1] and continuous mappings so from
is its usual elementary morphism.)
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The idea of this stronger bit of topology,
is that first of all, it makes it very intuitive
that it's built about neighboords, so that
the composition and decomposition,
vis-a-vis for example "Banach-Tarski decomposition",
results there's a different between decompositions
that result continuous domains degenerate decompositions
that don't, and that compositions with the non-contiguous
or not-neighborly, result naturally a piece-wise setting.
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Then also that it's already sort of attached to those
things already considered "continuous domains",
like lines and surfaces and manifolds and so on,
of the continuous sort.  I.e. it's usually "a continuous
topology is the hallmark topology of continuous domains".
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So, I encourage you to think this through,
as I am as well, to consider how it is that this
sort of "continuous topology" both leaves all
the existing theorems the same, and results new
ones that help correlate Lebesgue and Jordan measure,
and as with regards also to the three primary
continuous domains of line-reals, field-reals,
and signal-reals, as comprise a replete setting
of continuous domains and furthermore equip
about the minimal amount of the super-classical,
for line-reals, to result exactly a post-modern
paleo-classical, formalism in foundations for
mathematical continuity and infinity.
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Now, this doesn't really belong 500 posts deep
in a thread that's largely noise and less signal,
but it is a very strong bit of signal, and here as
that it's attached the overall larger linear narrative
of "A Theory" with its ubiquitous ordinals, an
extra-ordinary theory for 21'st century foundations.
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Ahem.
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