Re: Seven deadly sins of set theory

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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