Add S51 Khalimsky line, an example of a non-T1 cut point space

[danflapjax]

Space ID number 51 was unoccupied, so I decided to fill it with this space.

Five traits are currently unknown. Some comments on them:

• Contractible and Simply connected — Intuitively, I am fairly confident the space is contractible, but I haven't spent the time to write down anything explicitly. It is certainly simply connected.
• Door — This should be resolved by adding more theorems about door spaces, such as those in Introduce 'Almost discrete' property #821.
• Sober/Quasi-sober — This could be automatically derived by adding the theorem "locally finite implies quasi-sober", which follows from an easy generalization of this MathSE answer.

[yhx-12243]

One things noteworthy is that the usage of terminology of "irreducible", it is different from what it means in pi-base (P39).

[prabau]

One things noteworthy is that the usage of terminology of "irreducible", it is different from what it means in pi-base (P39).

I agree completely. We should not use that terminology here. See also my comments from Dec 20 in the last section of https://en.wikipedia.org/wiki/Talk:Cut_point, which I intend to clean up when I get the time.

It's debatable we even want to mention this here, as it is a true fact, but again, pi-base is not meant as a repository of all possible topology facts under the sun. Or is it? @StevenClontz

[danflapjax]

To me, the idea is that for any property P there is a corresponding property "irreducibly P" (c.f. "(weakly) locally P", "hereditarily P", etc.). If we care at all about such a notion—and I suspect we might given the recent interest in meta-properties—I think it is worth noting this fact. The terminology trouble in this case is that "cut-point space" is a noun phrase and so cannot be modified by an adverb, but maybe "irreducibly cut-point space" is just what we need to write.

[GeoffreySangston]

There was some discussion about this space beginning with @david20000813's comment #1020 (comment).

Note that 'digital line' is an alias for this when used in a subject called digital topology; (here's an arbitrary paper I found on Google Scholar which begins: 'The Khalimsky line or so called the digital line is the set $\mathbb{Z}$ of integers equipped with the topology $\kappa$ having $\mathcal{G}_\kappa := \{\{2m -1, 2m, 2m + 1\} | m \in \mathbb{Z}\}$ as a subbase... (followed by a bunch of references).')

[danflapjax]

Thanks for the links!

[StevenClontz]

pi-base is not meant as a repository of all possible topology facts under the sun. Or is it? @StevenClontz

Well one can write a computer program to generate any number of unique and true facts about general topology, so we will always need some threshold of editorial standards.

[prabau]

The change to P193, was that an end-of-line thing?

[danflapjax]

The change to P193, was that an end-of-line thing?

I assume so? I didn't intent to alter the file, but it showed up after I pulled in changes from main.

[prabau]

P205: Readers should not have to know what is meant by a "cutting". In other words, the justification should be based only on the pi-base definition of "cut point space".