Suppose that I have a finitely generated $\mathbb Z_p$-module $M$. I have read that there is a "$p$-adic topology" on it.
My questions are:
I have the product topology on the free group $\bigoplus_{i = 1}^n \mathbb Z_p$, and a homomorphism $\bigoplus_{i = 1}^n \mathbb Z_p \rightarrow M$. Is the $p$-adic topology the quotient topology on $M$?
Do I actually need the module $M$ to be finitely generated?