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Hometopological ring

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# topological ring

A ring $R$ which is a topological space is called a *topological ring* if the addition, multiplication, and the additive inverse functions are continuous functions from $R\times R$ to $R$.

A *topological division ring* is a topological ring such that the multiplicative inverse function is continuous away from $0$. A *topological field* is a topological division ring that is a field.

Remark. It is easy to see that if $R$ contains the multiplicative identity $1$, then $R$ is a topological ring iff addition and multiplication are continuous. This is true because the additive inverse of an element can be written as the product of the element and $-1$. However, if $R$ does not contain $1$, it is necessary to impose the continuity condition on the additive inverse operation.

## Mathematics Subject Classification

12J99*no label found*13J99

*no label found*54H13

*no label found*

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## Comments

## topological field

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?

## 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.