Property Suggestion: Locally orderable

[pzjp]

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):

• Circle and Long Circle
• S198
• S60
• S57
• Line with multiple origin S83, S84 and S85
• S79 and S87

Some examples not present in pi-Base are mentioned in RoML55-06 but probably not all of them are worth adding.

[prabau]

Maybe not the highest priority, but it looks like it would be interesting to add.

[prabau]

One minor request. Would it be possible to have zbmath links instead of MR links? Because zbmath is freely accessible, but MR is not.

[pzjp]

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.