pi-base
Property Suggestion: Brown space
Property Suggestion
A space is said to be Brown if for any two open sets $U$ and $V$, $\overline U\cap\overline V\ne\emptyset$.
Arguably, this is not the most important property in the world, but it may be nice to add. The property is used for example in various reasonings about the arithmetic sequence topologies.
Some spaces satisfying it: Golomb space (S52), Kirch space (S53), Irrational slope topology (S67).
A space with this property is not T2.5 (P4) in a strong sense. There are connections with various other properties, as given in the references below.
Some references: (1) Clark et al., A note on Golomb topologies (2019) https://zbmath.org/1420.54050 (available at https://www.researchgate.net/profile/Noah-Lebowitz-Lockard/publication/323367940_A_note_on_Golomb_topologies/links/5c24211792851c22a3484e7b/A-note-on-Golomb-topologies.pdf): See Proposition 7 and preceding definition.
(2) Alberto-Dominguez et al., "Totally Brown subsets of the Golomb space and the Kirch space" (https://zbmath.org/1563.11035) Section 3 has a lot of results relating it to other properties.
(3) Banakh & Stelmakh, "" (2023) https://zbmath.org/1546.54005 (https://arxiv.org/pdf/2211.12579) They define "Brown" a little differently: For every nonempty open sets $U and $V$, $U\cap V$ is infinite. See the discussion on p. 2 explaining when it coincides with the one from reference (1). We would have to explore what is most appropriate for pi-base.
Note: there is also an even stronger notion of totally Brown (= same condition for any finite number of nonempty open sets), called "superconnected" in an earlier paper of Banakh. See ref (2) why "superconnected" is not a good name. Anyway, if we decide to introduce Brown, I don't think we need to have "totally Brown" at this point.
So, do you think this property is worth introducing?
I may not have wanted to add this property in the past, I don't remember very well. Anyway, I agree it's not a very important property. On the other hand we have since added various other properties that did not seem that important either. It has close connections with other existing properties and may possibly help bookkeeping, making some asserted traits redundant.
So I mainly wanted to ask people's opinion about it. I am ok either way.
Let me expand a little on why I think this property should be added.
- We already have other analogous properties of such type. Indeed, Hausdorff and functionally Hausdorff spaces have their analogous anti-properties, namely anti-Hausdorff and strongly connected.
- The properties are elementary and in "the spirit" of pi-base. Indeed, pi-base includes a lot of properties which relate to separation properties, when they fail, and when they fail completely.
- I believe the properties are (perhaps without explicit mention) in Counterexamples anyway
- Better description of spaces. More properties means we describe them better.
- We have much more exotic properties that were added, and more exotic spaces. Adding such a simple to understand property should be a natural thing to do.
- The properties have clear potential in various theorems.
I agree with this. I searched it online and while "Brown space" is relatively obscure and most of the references being about Golomb/Kirch space, its still not totally random (unlike quasi-regular in the first reference you gave).
That being said, there are clearly quite big holes in pi-base in particular the properties in #818 as well as some other (like Artinian) should really be added, as they are actually used in reality.
