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 $$ \varprojlim_{i \in I} A_i \rightarrow A_j $$ 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.
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An inverse limit is a subspace of a product, and topologised as such. It's an initial topology itself by the associativity of initial topologies, using the restricted projections as you mention. This fact is mentioned at the end of my long post here that explains a lot of things about initial topologies (product topologies and subspace topologies are its best-known examples).
Henno Brandsma
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This was sort of thing that I could find when I looked up Wikipedia for example, but I was hoping for something without all the terminologies like initial topology ... – Johnny T. Nov 23 '17 at 07:59
\varprojlimformats as $\varprojlim$. (the name, I think, should be read as short for "variant: projective limit") – Nov 23 '17 at 20:11