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In this paper by Kaplansky on topological rings, in lemma 7 (reading the proof is not important to understanding my doubt), right ideal $B$ is given dense in $eA$ where $A$ is a topological ring (Hausdorff) and $e$ is an idempotent. Being dense, it means that any neighborhood around any point of $eA$ will contain a point of $B$. In this case we take $e\in A$, and the proof starts with saying that $B$ contains elements $e+x$ with $x$ arbitrary small, and so and so.

A similar thing was used in Lemma 2 where he chose $a$ sufficiently small, $x\in A$ and proved $a+x$ r.q.r. (definition of r.q.r. is not important to understand my doubt) and said $x$ has a neighborhood consisting of $r.q.r.$ elements.

Here is my doubt, what does he mean by $x$ and how can we choose it small. In topology (which doesn't have to be a metric), how can we talk about small numbers and also addition is only defined on the ring elements. $x$ must be in $A$ to be added with $e$. Also just by adding a small quantity in an element to make a very small neighborhood only works in metric spaces where distance is a thing. But what is the idea being used here for topological rings?

What concept am I missing here?

blabla
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2 Answers2

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Since $B$ is dense, it intersects every neighborhood of $e$. Since the neighborhoods of $e$ are of the form $e+U$, where $U$ is a neighborhood of $0$, we can say that, for each such $U$, there exists $x\in U$ such that $e+x\in B$.

“Small” means “belonging to a neighborhood of $0$”.

egreg
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We may perhaps view a statement

$S\subset A$ contains arbitrarily small elements.

as being a suggestive short-hand formulation for

There exists a sequence $s_n\to 0$ with $0\ne s_n\in S$.

Hagen von Eitzen
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