Property Suggestion: Has a ring/field topology

[felixpernegger]

Property Suggestion

A space is said to be "has a ring/field topology provided the topology is induced by a topological ring/field.
(I hope this hasnt been discussed before, didnt see anything)

Rationale

https://en.wikipedia.org/wiki/Topological_ring
https://en.wikipedia.org/wiki/Field_(mathematics)#Topological_fields

I haven't seen them mentioned explicitly before, but both topological rings (and to a lesser degree fields) seem to be widely studied. We of course have "has a group topology" so it seems natural to maybe extend this.
Of course one would need to be careful about what version of "ring" one is to use (although likely non-unital, noncommutative is the way to go).

Somewhat related, #1426 is an important property related to topological groups etc

Relationship to other properties

Obviously Field => Ring => Group. I haven't thought or read about more.
It seems to me if we want this property or not mainly depends on if we can find more theorems.

[prabau]

Pi-base classifies spaces up to homeomorphisms. So we don't really care about topological groups per se (there can be many structures of a topological group on a space with the same underlying topological space). But the fact that a space is "groupable" (horrible word, I know), i.e., there there exists a group structure on the underlying space that makes it a topological group, is important because it imposes all sorts of constraints on the topology itself.

Similarly, we wouldn't care about topological rings per se. From the fact that there is a ring structure on a topological space with continuous ring operations, can we infer things on the space itself that could not already be inferred from the fact that it's also a topological group? I think that's the important question.
If we cannot answer that very satisfactorily, I am not sure we need to add this property.

[felixpernegger]

yes this is what i meant earlier. I suspect there might be stuff (especially for fields), but what exactly is to be seen

[Moniker1998]

Of course one would need to be careful about what version of "ring" one is to use (although likely non-unital, noncommutative is the way to go).

This is pretty much why I would be keen to not add this