topological ring

Primary tabs

Defines:
topological field, topological division ring
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

Mathematics Subject Classification

topological field

Is it not necessary for a topological field
to have a continuous inverse operation, just
like for a topological group?

that it is a field and a topological ring?

Re: topological field

I am not sure. This is one item that I will have to investigate further.

For starters, continuity at 0 cannot be expected at all: x -> 1/x is almost never continuous at 0, yet we would like fields such as R and C to be topological fields.

The books that I have looked at do not specify in the definition that multiplicative inverse or additive inverse operations have to be continuous in a topological field or ring. Unfortunately they also do not address the issue of whether such continuity is automatically implied by the definition or not.

Re: topological field

The additive inverse operation is continuous in every topological ring since -x = (-1)*x.

But the map x -> x^(-1) need not be continuous, so for topological fields one has to require that separately.