TITLE: Inverse limit of cohomology of direct system of topological spaces induces an isomorphism in cohomology.
QUESTION [5 upvotes]: Let $X_n$ be a direct system of topological spaces approximating $X$. Suppose $H^k(X_n)\cong H^k(X)$ for $n>k$. I want to show that there is a ring isomorphism
$$H^*(X)\rightarrow \text{lim}_{\leftarrow}H^*(X_n)$$
The direct system of topological spaces induces an inverse limit of rings. I can beleive that this would hold degree wise since the $H^k(X_n)$ stabilize for large $n$ (although im not actually sure how to prove this). But I don't see at all why this should be an isomorphism of rings, its not clear to me at all how the multiplication structure comes into this.

REPLY [4 votes]: Does the map $X_n \rightarrow X$ induce an isomorphism $H^k(X) \rightarrow H^k(X_n)$ if $n > k$?
In that case it's pretty trivial. If you have two diagrams $D,C: \mathbb{N} \rightarrow \mathcal C$ and a natural transformation $\phi: C \rightarrow D$ then if for some $k$ we have that $\phi_n: C_n \rightarrow D_n$ is an isomorphism for all $n > k$, $\text{colim}(\phi): \text{colim}(C) \rightarrow \text{colim}(D)$ (the inverse limit is a special case of the colimit) is an isomorphism. In your case $C$ is the constant diagram $H^k(X)$ and $D$ is $H^k(X_n)$. No facts about ring structure needed. Just abstract nonsense.