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topological ring

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

Mathematics Subject Classification

12J99 no label found13J99 no label found54H13 no label found

Comments

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

Or does this already follow from the facts
that it is a field and a topological ring?

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.

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.

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