# Definition of the topology on the inverse limit

I just want to make sure that I have the topology on the inverse limit correctly as I am having quite a bit of difficulty understanding this. If $\lim_{\leftarrow i \in I} A_i$ is some inverse limit then the topology on it is the coarsest collection of open sets such that the projection $$\lim_{\leftarrow i \in I} A_i \rightarrow A_j$$ is continuous for each $j \in I$. This is my understanding. Could I possibly verify this with someone because I just can't find anywhere that spells it out.

• Yes, that's right, same as the product topology. – Qiaochu Yuan Nov 22 '17 at 22:47
• @QiaochuYuan Good to know! Thanks! – Johnny T. Nov 22 '17 at 23:03
• Aside: \varprojlim formats as $\varprojlim$. (the name, I think, should be read as short for "variant: projective limit") – user14972 Nov 23 '17 at 20:11