Add metric fan

[StevenClontz]

See discussion at #803

[danflapjax]

Some easy properties to add:

• P181 Countably infinite
• P126 Door, as the space refines S12

[prabau]

Just curious about the terminology "sequential fan" and "metric fan". I would have thought "sequential fan" is because each of the "spines" is itself (homeomorphic to) a convergent sequence. But the "metric fan" has the same property: each spine is also a convergent sequence. we could say it's a "metric sequential fan", versus the other one, which is not metrizable.

Apart from that article from Gruenhage, what is the more commonly used terminology for metric vs non-metric fans? There could also be similar "fans" with spines being sequences or being something else? Hedgehogs?

[prabau]

About having "Compact semi-Hausdorff non-KC space" as an alias, I can't think of anyone referencing this space by such an unhelpful and nondescript name. I don't think it helps anyone to keep that name. And I don't imagine anyone having referred to this space by that name, and definitely not from the literature. (And nobody will care if we remove this alias.)

[StevenClontz]

Just curious about the terminology "sequential fan" and "metric fan". I would have thought "sequential fan" is because each of the "spines" is itself (homeomorphic to) a convergent sequence. But the "metric fan" has the same property: each spine is also a convergent sequence. we could say it's a "metric sequential fan", versus the other one, which is not metrizable.

The "sequential fan" name comes from the fact that open neighborhoods of the non-isolated point are defined by sequences $\omega\to\omega$, rather than elements $n<\omega$ as in the metric fan (named because it's metrizable, ofc).

Apart from that article from Gruenhage, what is the more commonly used terminology for metric vs non-metric fans? There could also be similar "fans" with spines being sequences or being something else? Hedgehogs?

I'm working off memory, but "metric fan" and "sequential fan" are standard names (at least within the southeast US set-theoretic topology colloquia). I do think I've heard "hedgehog" before as well, though that might be used to distinguish using [0,1] in place of \omega+1: https://en.wikipedia.org/wiki/Hedgehog_space

[prabau]

FYI, see https://mathoverflow.net/a/370989. At the end of the post, T. Banakh gives the "metric fan" with the same definition as you. He also mentions the "Frechet-Urysohn fan" at the end. I have not looked into it, but that could be interesting too; or is that the same as the "sequential fan"? (See also the linked paper)

[StevenClontz]

I think it's the sequential fan: it's an example of https://topology.pi-base.org/spaces/S000131/properties/P000080 that's not first countable (or even a W space).

[prabau]

Cool. I'll let @danflapjax do the review.