Left-ray topology on $\omega_1$ - #1591

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Left-ray topology on $\omega_1$#1591
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The latest addition in the Artinian saga.

See #1589.

We could add some more traits in this PR or later.

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I don't plan on working on this or my other PR's anytime soon anymore, so I close them for now.

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prabau marked this pull request as draft January 11, 2026 22:57
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@prabau since I saw you changed the definition, I copied the definition from S199 (left ray topology on $\omega$), so if you prefer this, maybe we should also change it there

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One of the reasons I changed the definition is that the previous one had a problem. It claimed the collection of open sets was (apart from boundary cases) exactly all the "closed rays", and also was exactly all the "open rays". That does not quite work, due to limit ordinals below $\omega_1$.

The new description (inspired by S166) also seemed more informative and easier to grasp.

Not sure I want to modify S199 here, but it could be helpful.

felixpernegger
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Comment threadspaces/S000217/properties/P000093.md

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I dont really like this, since the "basic open set" depends on a basis, but the basis isn't fixed or anything.

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You are right. It was referring to the base described in the definition, but it's out of context here.
I'll modify.

@felixpernegger

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@prabau I think all your changes are good now, if you are happy with all my (initial) files, I guess feel free to apporve this.

This space can easily be completed, but maybe its more appropriate to do this later.

@prabau
prabau merged commit cdfae02 into mainJan 16, 2026
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prabau deleted the uncountable-left-ray branch January 16, 2026 23:43
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Left-ray topology on $\omega_1$ - #1591
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Left-ray topology on $\omega_1$#1591
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The latest addition in the Artinian saga.
See #1589.
We could add some more traits in this PR or later.
@felixpernegger
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I don't plan on working on this or my other PR's anytime soon anymore, so I close them for now.

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@prabau since I saw you changed the definition, I copied the definition from S199 (left ray topology on $\omega$), so if you prefer this, maybe we should also change it there
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One of the reasons I changed the definition is that the previous one had a problem. It claimed the collection of open sets was (apart from boundary cases) exactly all the "closed rays", and also was exactly all the "open rays". That does not quite work, due to limit ordinals below $\omega_1$.
The new description (inspired by S166) also seemed more informative and easier to grasp.
Not sure I want to modify S199 here, but it could be helpful.
felixpernegger
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I dont really like this, since the "basic open set" depends on a basis, but the basis isn't fixed or anything.
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You are right. It was referring to the base described in the definition, but it's out of context here.

I'll modify.
@felixpernegger
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@prabau I think all your changes are good now, if you are happy with all my (initial) files, I guess feel free to apporve this.
This space can easily be completed, but maybe its more appropriate to do this later.
@prabau
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merged commit cdfae02
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Left-ray topology on $\omega_1$ - #1591
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prabau merged 5 commits into
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Merged
Left-ray topology on $\omega_1$#1591
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uncountable-left-ray
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@felixpernegger
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The latest addition in the Artinian saga.
See #1589.
We could add some more traits in this PR or later.
@felixpernegger
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Jan 10, 2026

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I don't plan on working on this or my other PR's anytime soon anymore, so I close them for now.

This was referenced Jan 10, 2026
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@prabau since I saw you changed the definition, I copied the definition from S199 (left ray topology on $\omega$), so if you prefer this, maybe we should also change it there
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One of the reasons I changed the definition is that the previous one had a problem. It claimed the collection of open sets was (apart from boundary cases) exactly all the "closed rays", and also was exactly all the "open rays". That does not quite work, due to limit ordinals below $\omega_1$.
The new description (inspired by S166) also seemed more informative and easier to grasp.
Not sure I want to modify S199 here, but it could be helpful.
felixpernegger
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Comment threadspaces/S000217/properties/P000093.md
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I dont really like this, since the "basic open set" depends on a basis, but the basis isn't fixed or anything.
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You are right. It was referring to the base described in the definition, but it's out of context here.

I'll modify.
@felixpernegger
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@prabau I think all your changes are good now, if you are happy with all my (initial) files, I guess feel free to apporve this.
This space can easily be completed, but maybe its more appropriate to do this later.
@prabau
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merged commit cdfae02
into
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Left-ray topology on $\omega_1$ - #1591
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prabau merged 5 commits into
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Merged
Left-ray topology on $\omega_1$#1591
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The latest addition in the Artinian saga.
See #1589.
We could add some more traits in this PR or later.
@felixpernegger
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I don't plan on working on this or my other PR's anytime soon anymore, so I close them for now.

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@prabau since I saw you changed the definition, I copied the definition from S199 (left ray topology on $\omega$), so if you prefer this, maybe we should also change it there
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One of the reasons I changed the definition is that the previous one had a problem. It claimed the collection of open sets was (apart from boundary cases) exactly all the "closed rays", and also was exactly all the "open rays". That does not quite work, due to limit ordinals below $\omega_1$.
The new description (inspired by S166) also seemed more informative and easier to grasp.
Not sure I want to modify S199 here, but it could be helpful.
felixpernegger
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Comment threadspaces/S000217/properties/P000093.md
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I dont really like this, since the "basic open set" depends on a basis, but the basis isn't fixed or anything.
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You are right. It was referring to the base described in the definition, but it's out of context here.

I'll modify.
@felixpernegger
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@prabau I think all your changes are good now, if you are happy with all my (initial) files, I guess feel free to apporve this.
This space can easily be completed, but maybe its more appropriate to do this later.
@prabau
prabau
merged commit cdfae02
into
mainJan 16, 2026

1 check passed

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Left-ray topology on $\omega_1$ - #1591
Merged
prabau merged 5 commits into
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uncountable-left-ray
Jan 16, 2026
Merged
Left-ray topology on $\omega_1$#1591
prabau merged 5 commits into
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uncountable-left-ray
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The latest addition in the Artinian saga.
See #1589.
We could add some more traits in this PR or later.
@felixpernegger
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Jan 10, 2026

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I don't plan on working on this or my other PR's anytime soon anymore, so I close them for now.

This was referenced Jan 10, 2026
 Merged

 Closed

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@prabau
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@prabau since I saw you changed the definition, I copied the definition from S199 (left ray topology on $\omega$), so if you prefer this, maybe we should also change it there
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One of the reasons I changed the definition is that the previous one had a problem. It claimed the collection of open sets was (apart from boundary cases) exactly all the "closed rays", and also was exactly all the "open rays". That does not quite work, due to limit ordinals below $\omega_1$.
The new description (inspired by S166) also seemed more informative and easier to grasp.
Not sure I want to modify S199 here, but it could be helpful.
felixpernegger
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commented
Jan 16, 2026
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Comment threadspaces/S000217/properties/P000093.md
@felixperneggerfelixperneggerJan 16, 2026

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I dont really like this, since the "basic open set" depends on a basis, but the basis isn't fixed or anything.
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You are right. It was referring to the base described in the definition, but it's out of context here.

I'll modify.
@felixpernegger
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Jan 16, 2026

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@prabau I think all your changes are good now, if you are happy with all my (initial) files, I guess feel free to apporve this.
This space can easily be completed, but maybe its more appropriate to do this later.
@prabau
prabau
merged commit cdfae02
into
mainJan 16, 2026

1 check passed

@prabau
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Skip to content
Left-ray topology on $\omega_1$ - #1591
Merged
prabau merged 5 commits into
mainfrom 
uncountable-left-ray
Jan 16, 2026
Merged
Left-ray topology on $\omega_1$#1591
prabau merged 5 commits into
mainfrom 
uncountable-left-ray
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The latest addition in the Artinian saga.
See #1589.
We could add some more traits in this PR or later.
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Jan 10, 2026

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I don't plan on working on this or my other PR's anytime soon anymore, so I close them for now.

This was referenced Jan 10, 2026
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 Closed

@prabauprabau
reopened this
Jan 11, 2026
@prabau
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marked this pull request as draft
January 11, 2026 22:57
@prabau
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marked this pull request as ready for review
January 16, 2026 02:27
felixpernegger
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Comment threadspaces/S000217/README.md
@felixperneggerfelixperneggerJan 16, 2026

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@prabau since I saw you changed the definition, I copied the definition from S199 (left ray topology on $\omega$), so if you prefer this, maybe we should also change it there
@prabauprabauJan 16, 2026

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One of the reasons I changed the definition is that the previous one had a problem. It claimed the collection of open sets was (apart from boundary cases) exactly all the "closed rays", and also was exactly all the "open rays". That does not quite work, due to limit ordinals below $\omega_1$.
The new description (inspired by S166) also seemed more informative and easier to grasp.
Not sure I want to modify S199 here, but it could be helpful.
felixpernegger
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Jan 16, 2026
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Comment threadspaces/S000217/properties/P000093.md
@felixperneggerfelixperneggerJan 16, 2026

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I dont really like this, since the "basic open set" depends on a basis, but the basis isn't fixed or anything.
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You are right. It was referring to the base described in the definition, but it's out of context here.

I'll modify.
@felixpernegger
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Jan 16, 2026

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@prabau I think all your changes are good now, if you are happy with all my (initial) files, I guess feel free to apporve this.
This space can easily be completed, but maybe its more appropriate to do this later.
@prabau
prabau
merged commit cdfae02
into
mainJan 16, 2026

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January 16, 2026 23:43
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Skip to content
Left-ray topology on $\omega_1$ - #1591
Merged
prabau merged 5 commits into
mainfrom 
uncountable-left-ray
Jan 16, 2026
Merged
Left-ray topology on $\omega_1$#1591
prabau merged 5 commits into
mainfrom 
uncountable-left-ray
Conversation
@felixpernegger
@felixperneggerfelixpernegger
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Jan 9, 2026

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The latest addition in the Artinian saga.
See #1589.
We could add some more traits in this PR or later.
@felixpernegger
felixpernegger
commented
Jan 10, 2026

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I don't plan on working on this or my other PR's anytime soon anymore, so I close them for now.

This was referenced Jan 10, 2026
 Merged

 Closed

@prabauprabau
reopened this
Jan 11, 2026
@prabau
prabau
marked this pull request as draft
January 11, 2026 22:57
@prabau
prabau
marked this pull request as ready for review
January 16, 2026 02:27
felixpernegger
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Jan 16, 2026
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Comment threadspaces/S000217/README.md
@felixperneggerfelixperneggerJan 16, 2026

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@prabau since I saw you changed the definition, I copied the definition from S199 (left ray topology on $\omega$), so if you prefer this, maybe we should also change it there
@prabauprabauJan 16, 2026

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One of the reasons I changed the definition is that the previous one had a problem. It claimed the collection of open sets was (apart from boundary cases) exactly all the "closed rays", and also was exactly all the "open rays". That does not quite work, due to limit ordinals below $\omega_1$.
The new description (inspired by S166) also seemed more informative and easier to grasp.
Not sure I want to modify S199 here, but it could be helpful.
felixpernegger
felixpernegger
commented
Jan 16, 2026
View reviewed changes
Comment threadspaces/S000217/properties/P000093.md
@felixperneggerfelixperneggerJan 16, 2026

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I dont really like this, since the "basic open set" depends on a basis, but the basis isn't fixed or anything.
@prabauprabauJan 16, 2026

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You are right. It was referring to the base described in the definition, but it's out of context here.

I'll modify.
@felixpernegger
felixpernegger
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Jan 16, 2026

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@prabau I think all your changes are good now, if you are happy with all my (initial) files, I guess feel free to apporve this.
This space can easily be completed, but maybe its more appropriate to do this later.
@prabau
prabau
merged commit cdfae02
into
mainJan 16, 2026

1 check passed

@prabau
prabau
deleted the

uncountable-left-ray

branch
January 16, 2026 23:43
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, 'i');
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})();

Skip to content
Left-ray topology on $\omega_1$ - #1591
Merged
prabau merged 5 commits into
mainfrom 
uncountable-left-ray
Jan 16, 2026
Merged
Left-ray topology on $\omega_1$#1591
prabau merged 5 commits into
mainfrom 
uncountable-left-ray
Conversation
@felixpernegger
@felixperneggerfelixpernegger
commented
Jan 9, 2026

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The latest addition in the Artinian saga.
See #1589.
We could add some more traits in this PR or later.
@felixpernegger
felixpernegger
commented
Jan 10, 2026

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CollaboratorAuthor
I don't plan on working on this or my other PR's anytime soon anymore, so I close them for now.

This was referenced Jan 10, 2026
 Merged

 Closed

@prabauprabau
reopened this
Jan 11, 2026
@prabau
prabau
marked this pull request as draft
January 11, 2026 22:57
@prabau
prabau
marked this pull request as ready for review
January 16, 2026 02:27
felixpernegger
felixpernegger
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Jan 16, 2026
View reviewed changes
Comment threadspaces/S000217/README.md
@felixperneggerfelixperneggerJan 16, 2026

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@prabau since I saw you changed the definition, I copied the definition from S199 (left ray topology on $\omega$), so if you prefer this, maybe we should also change it there
@prabauprabauJan 16, 2026

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One of the reasons I changed the definition is that the previous one had a problem. It claimed the collection of open sets was (apart from boundary cases) exactly all the "closed rays", and also was exactly all the "open rays". That does not quite work, due to limit ordinals below $\omega_1$.
The new description (inspired by S166) also seemed more informative and easier to grasp.
Not sure I want to modify S199 here, but it could be helpful.
felixpernegger
felixpernegger
commented
Jan 16, 2026
View reviewed changes
Comment threadspaces/S000217/properties/P000093.md
@felixperneggerfelixperneggerJan 16, 2026

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I dont really like this, since the "basic open set" depends on a basis, but the basis isn't fixed or anything.
@prabauprabauJan 16, 2026

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You are right. It was referring to the base described in the definition, but it's out of context here.

I'll modify.
@felixpernegger
felixpernegger
commented
Jan 16, 2026

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CollaboratorAuthor
@prabau I think all your changes are good now, if you are happy with all my (initial) files, I guess feel free to apporve this.
This space can easily be completed, but maybe its more appropriate to do this later.
@prabau
prabau
merged commit cdfae02
into
mainJan 16, 2026

1 check passed

@prabau
prabau
deleted the

uncountable-left-ray

branch
January 16, 2026 23:43
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