Remark

Denote the endomorphism ring of the additive group of M by E. Endow M with the discrete topology; the ring E consists of mappings from M to M, and we can endow it with the topology induced by the product topology on $M^M$ (“topology of simple convergence in M”, Gen. Top., I, §2, No. 3, p. 31). The topology on E is Hausdorff and compatible with the additive group structure on E. For every $f$ in E, the mappings $g\mapsto f\circ g$ and $g\mapsto g\circ f$ from E to E are continuous. Consequently, the commutant of every subset of E is closed in E. Theorem 3 therefore implies that $A''_M$ is the closure of $A_M$ in E.