Exercise 17, § 2

It is said that on an $A$-module $E$ a linear topology $\mathscr{C}$ is minimal if it is separated and if there does not exist a separated linear topology strictly less fine than $\mathscr{C}$.

a) For a separated linear topology $\mathscr{C}$ on $E$ to be minimal it is necessary and sufficient that every filter base $\mathscr{B}$ on $E$, formed by affine linear varieties, and having a single adherent point, converge 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 $0$ formed by submodules, the $M + V$ form a filter base having the same adherent points as $\mathscr{B}$. To see that the condition is sufficient, remark that a filter base formed by open submodules, whose intersection is reduced to $0$, is a fundamental system of neighbourhoods of $0$ for a separated linear topology less fine than $\mathscr{C}$.)

b) For the discrete topology on an $A$-module $E$ to be minimal it is necessary and sufficient that $E$ be artinian.

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