What is the local basis at $0$ of inverse limit topology?

Question

What is the local basis of inverse limit topology at $0$?

For example, $\mathbb Z_p＝\lim\mathbb Z/p^n\mathbb Z$ has $$\{ \{(\cdots,α_0)\in\mathbb Z_p｜α_m＝・・・＝α_o＝0\} \mid m≧0\}$$as a local basis at $0$ ?

Self contained explanation or reference(website only) are both appreciated, thank you in advance.

Answer 1

If $R_n$ is a sequence of topological groups or rings, with maps $\rho_n:\mathbb R_{n+1}\to\mathbb R_n,$ then if $R$ is the inverse limit, with maps $\phi_n:R\to R_n,$ we get a local basis for $R$ at $0$ in terms of a local basis for each $R_n$ at $0.$

If $B_n$ is a local basis of $R_n$ at $0,$ then a local basis for $R$ can be defined at:

$$\{\phi_n^{-1}(U)\mid n\in\mathbb N,U\in B_n\}$$

Of course, if each $R_n$ is discrete, we can take $B_n=\{\{0\}\},$ and then the local basis of $R$ consists of the kernels of the $\phi_n,$ which is your local basis for $p$-adics.

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The reason for this is that $R$ can be seen as a sub-topological ring of $\prod_n R_n.$ In particular, it inherits the product topology.