pi-base
Property Suggestion: Locally orderable
Property Suggestion
A space is said to be Locally orderable (Locally ordered topological space) provided it admits an open cover consisting of LOTS P133. According to pi-Base convention this should be called weakly locally orderable (or weakly locally LOTS) but both definitions are in this case equivalent (every LOTS is not-weakly locally orderable) so we shall omit the weakness.
Rationale
This property is very natural, but apparently was studied in MR0154246 (1962) and MR4183420/Zbl 1458.54023 (2020) only.
Relationship to other properties
For (my own) convenience, below I will use Polish abbreviation LPPT for locally ordered topological space (pol. lokalnie porządkowa przestrzeń topologiczna).
- Every LOTS is LPPT
- Every LPPT is Locally Hausdorff P84.
- LPPT + $T_2$ P3+ Locally connected P41 => Weakly locally compact P23.
- LPPT + Regular P11 => Tychonoff P5
- LPPT + P11 + Connected P36 => Locally connected P41
- LPPT + Paracompact P30 + $T_2$ P3 => $T_5$ P8
Some locally orderable spaces on pi-Base (except LOTS):
Some examples not present in pi-Base are mentioned in RoML55-06 but probably not all of them are worth adding.
Hereditarity:
Local orderability is hereditary on open subspaces. Under the assumption of $T_2$ it is hereditary on compact subsets. Under the assumption of $T_3$ it is hereditary on connected subsets.
Maybe not the highest priority, but it looks like it would be interesting to add.
One minor request. Would it be possible to have zbmath links instead of MR links? Because zbmath is freely accessible, but MR is not.
MR4183420 is Zbl 1458.54023 and is open access doi MR0154246 is the PhD dissertation by Horst Herrlich (Ordnungsfähigkeit topologischer Räume) not indexed on zbMATH and is hardly accessible anyway.
I started experimenting with this property on a new branch locally-orderable. There are loads of spaces requiring decing local orderability by hand. For now I used free ids P120 and T750+
Great. One request. Please do NOT create a gigantic PR for this. The best way to have this moving forward is to create a few smaller PR that can be reviewed more quickly and get each one merged first before doing the next one.
I would do: first one PR with the definition and a few theorems (or more theorems if it's not too involved, but if there are too many theorems, they can be done in more than one PR). Then other PRs with traits for specific spaces, cleanup of existing related traits, etc. (split these into multiple PRs if necessary).
Keeping this manageable will really help this moving forward, and it allows to experiment in pi-base after each stage.
So possibly the current stage is good for the first PR. It's good idea to merge it relatively soon, before someone uses the same ids...
It would take significant time to add the trait to all spaces. Actually I will not proceed too far with it anyway. I believe some additional theorems could be provided to decide the trait, even though I do not have those at hand.
It's more than enough. It would have been better to only have the definition and some theorems, without any traits for specific spaces. Can you remove the traits and add them in a separate PR later on?
I saw that metrization theorem for locally orderable spaces (2.12 in the article) would resolve few traits in pi-base: Locally orderable + G-delta diagonal + Paracompact +Hausdorff => Metrizable
All assumptions are essential due to the respective counterexamples: S21, S36, S56, S83
The theorem would not help establishing metrizability but several cases of no local orderability can be detected.
Actually we should probably add just
Locally orderable + $G_\delta$ diagonal $\Rightarrow$ Locally metrizable
And a hereditarity (meta-property) to G-delta diagonal.
I guess we can close this. Possibly adding some new examples may be another issue.