Exercise 17, § 2

A linear topology $\mathscr{T}$ on an A-module E is called minimal if it is Hausdorff and there exists no Hausdorff linear topology strictly coarser than $\mathscr{T}$.

(a) For a Hausdorff linear topology $\mathscr{T}$ on E to be minimal, it is necessary and sufficient that every filter base $\mathscr{B}$ on E, consisting of affine linear varieties and having a single cluster point, be convergent to this point. (To see that the condition is necessary, observe that, when M runs through $\mathscr{B}$ and V a fundamental system of neighbourhoods of 0 consisting of submodules, the $M + V$ form a filter base with the same cluster points as $\mathscr{B}$. To see that the condition is sufficient, note that a filter base consisting of open submodules, whose intersection is reduced to 0, is a fundamental system of neighbourhoods of 0 in a Hausdorff linear topology coarser than $\mathscr{T}$.)

(b) For a discrete topology on an A-module E to be minimal, it is necessary and sufficient that E be Artinian.

(c) If $\mathscr{T}$ is minimal, the topology induced by $\mathscr{T}$ on any closed submodule of E is minimal.