I am currently working on the infinite-dimensional case of the Atiyah-Hirzebruch spectral sequence where a lot of inverse limits are needed to state useful conditions on convergence.
In particular let $X$ be a CW-Complex, $E$ a CW-Spectrum and $E*$ the induced generalized cohomology theory.
Then what does $$ \varprojlim E^*(X,X_n)=0$$ mean? Is there a way to intuitively understand inverse limits being zero?
If one had an inverse System $(E^k,d_k)_k$ in an abelian Category with $E^{k+1}=E^k$ for all $k>r$ would the inverse limit be $0$?