Exercise 15, § 2

(a) Let $E$ be a linearly topologized A-module (Exercise 14). For a filter base $\mathfrak{B}$ on $E$ consisting of linear varieties to admit a cluster point, it is necessary and sufficient that there exists a convergent filter base $\mathfrak{B}' \supset \mathfrak{B}$ consisting of linear affine varieties. $E$ is said to be linearly compact if it is Hausdorff and every filter base on $E$ consisting of affine linear varieties admits at least one cluster point. Every Artinian module is linearly compact with the discrete topology. Every linearly compact submodule of a Hausdorff linearly topologized module $F$ is closed in $F$.

(b) If $E$ is a linearly compact A-module and $u$ a continuous linear mapping of $E$ to a Hausdorff linearly topologized A-module $F$, $u(F)$ is a linearly compact submodule of $F$.

(c) Let $E$ be a Hausdorff linearly topologized A-module and $F$ a closed submodule of $E$. For $E$ to be linearly compact, it is necessary and sufficient that $F$ and $E/F$ be so.

(d) Every product of linearly compact modules is linearly compact (consider a filter base which is maximal among the filter bases consisting of affine linear varieties.

(e) Let $(E_\alpha, f_{\alpha\beta})$ be an inverse system of linearly topologized modules relative to a directed indexing set; suppose that the $f_{\alpha\beta}$ are continuous linear mappings and that, for $\alpha \leq \beta$, $f_{\alpha\beta}^{-1}(0)$ is a linearly compact submodule of $E_\alpha$. Let $E = \lim_{\leftarrow} E_\alpha$, and let $f_\alpha$ be the canonical mapping $E \to E_\alpha$; show that, for all $\alpha$, $f_\alpha(E) = \bigcap_{\alpha \leq \beta} f_{\alpha\beta}(E_\beta)$ (in particular, if the $f_{\alpha\beta}$ are surjective, so are the $f_\alpha$) and $f_\alpha^{-1}(0)$ is linearly compact (use (d) and General Topology, Chapter I, Appendix, no. 2, Theorem 1).