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Let $M$ be an $R$-module ($R$ commutative ring with unity). Let $M=M_0 \supseteq M_1\supseteq M_2\supseteq\cdots$ be a chain of submodules. The topology in $M$: The open sets in $M$ are arbitrary union of the sets of the form $m+M_n$. My question is with respect to this topology why the addition map on $M$ is continuous? Thank you in advance.

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The inverse image of $m+M_n$ w.r.t. the addition map is $$\bigcup_{a \in M} \Bigg (\Big( a + M_n \Big) \times \Big((m-a) + M_n \Big) \Bigg),$$ which is open.

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