Let \(R\) be an associative ring. The set \(\text{Irr\,}R\) of isomorphism classes of irreducible modules is provided with a refined Zariski topology which is defined as follows. A subset \(X\) of \(\text{Irr\,}R\) is closed if \(X\) is closed under simple subquotients of direct products of elements from \(X\). If \(I\) is an ideal in \(R\) denote by \(V(I)\) the set of all elements from \(\text{Irr\,}R\) that are annihilated by the ideal \(I\). Suppose that \(R\) has the ascending chain condition on semiprimitive ideals.
It is shown that each closed subset of \(\text{Irr\,}R\) has the form \(V(I)\cup S\), where \(I\) is a semiprime ideal in \(R\) and \(S\) is a finite subset in \(\text{Irr\,}R\) if one of the following conditions is satisfied: \(R\) has either a special cofinite product condition on its semiprimitive prime ideals which is satisfied if \(R\) is an algebra of countable dimension or \(R\) has countable left Krull dimension. In all cases the refined Zariski topology on \(\text{Irr\,}R\) coincides with the closed-point Zariski topology and it is Noetherian.