Exercise 27, § 6

Let $A$ be a separated integral topological ring, $A'$ the set of elements $\neq 0$ of $A$, $R$ the equivalence relation $xt = yz$ between the pairs $(x, y)$ and $(z, t)$ in $A \times A'$; the quotient set $K = (A \times A')/R$ is the field of fractions of $A$.

a) Endow $A'$ with the topology induced by that of $A$, $A \times A'$ with the product topology. Prove that if the relation $R$ is open, the quotient topology of that of $A \times A'$ by $R$ is a separated topology on $K$, compatible with the field structure of $K$.

b) Suppose that, for every $a \neq 0$ in $A$, the mapping $x \mapsto ax$ of $A$ into itself is open. Prove that $R$ is then open.

c) Suppose that $K$ is endowed with a topology $\mathfrak{T}$ compatible with its ring structure and inducing on $A$ (canonically embedded in $K$) the given topology. Deduce from a) and b) that there exists on $K$ a separated topology $\mathfrak{T}'$, less fine than $\mathfrak{T}$ and compatible with the field structure of $K$.