inverse system vs inverse sequence

Question

I am wondering about such problem. Let $\{X_i,\phi_{ij},I\}$ be an inverse system, where the directed set $I$ has such property that there exists a sequence $i_1 \leq i_2\leq\cdots\subset I$ such that for every $j\in I$ there exists $i_n$ with $i_n\geq j$.

So it gives us inverse sequence indexed by the elements of the sequence. Are there any connections between inverse limits of these inverse systems? Maybe they are the same sometimes? I am especially interested in inverse limits of topological spaces.

Answer 1

Write the inverse limit of the full system as $A$ and of the subsystem as $B$. We have a map from $A$ to $B$ given on $X_{i_j}$ by the composition of the identity on $X_{i_j}$ with the projection of $A$ onto $X_{i_j}$ and a map $B\to A$ given similarly on the $X_{i_j}$ and on the other $X_i$ by composing the identity on $X_i$ with the given map $X_{i_n}\to X_i$ and the projection $A\to X_{i_n}$. You can check directly that both compositions are the identity, since their components are the projections $A\to X_i$ and $B\to X_{i_j}$.

Answer 2

The sequence is also directed, and cofinal in I. It's a theorem that the inverse limits of the inverse systems are isomorphic.