# Questions tagged [topological-rings]

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### A weird connection between Topology and algebra

It is common for mathematicians to try and merge different structures which they find interesting. Topological groups and Topological rings are examples of this, where one defines a topology and a ...
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### A hyper-algebraic characterisation of topological rings

This is somewhat of a follow up to this question. Background: I am interested in topological rings, and particularly in unifying the construction of the $I$-adic Topology in an arbitrary ring, with ...
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### Sufficient condition for multiplication to be continuous

Given a commutative ring $(R,+,\cdot)$, and a topology $\tau$ on $R$ such that for any $a\in R$ the maps \begin{align} \cdot a: &R\to R \\ &x\mapsto a\cdot x \\\\ &\mbox{and} \\\\ +a: &...
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### A natural topology on a field

I can endow any field with a natural topology in the following way. Given a polynomial $f\in K[X]$, I denote by $\mathcal{O}(f)=\{x\in K\mid\exists y\in K^{\times}\ f(x)=y^2\}$, i.e. the set of ...
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### Unique vector space topology on $F^n$?

Every finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$ has a unique topology that makes addition and scalar multiplication continuous. Is the same true of finite dimensional vector ...
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### When can monothetic groups be turned into rings?

If G is a Hausdorff topological group, saying that G is monothetic is equivalent to saying there exists a homomorphism $f: \mathbb{Z} \to G$ with dense image. A multiplication can be naturally defined ...
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### Closure of subalgebras of $C(X,\Bbb R)$.

I attempted to give a generalization of Stone-Weierstrass Theorem below, but I am not sure if it is correct. If it is not correct, I would like to have a counterexample. Let $X$ be a compact ...
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