What does go wrong in Cellular homology if one considers projective limits of celullar complexes instead of CW-complexes?

Consider a nice topological space $X$ (e.g. the 3-sphere) and consider inside a decreasing sequence of compact subsets $(K_n)_{ninmathbb N}$ such that $K_infty:=bigcap_{nin mathbb N} K_n$ is 0-dimensional. Assume further more that you have a filtration of CW-complexes $(mathcal S_n)_{nin mathbb N}$, each on $Xsetminus Int(K_n)$ for all $nin mathbb N$. Now consider the projective limit of the system
$$ X/K_{n+1} xrightarrow{pi_{n+1,n}} X/K_n$$
where $X/K_n := K/sim$ with $xsim y$ identifying elements of the same connected components of $K_n$. We can thus define $X’:=varprojlim X/K_n$ which comes with a natural map $Xrightarrow X’$. Assuming $X$ and the $K_n$ are nice enough (like $X$ compact and for any neighborhood $mathcal U$ of any given $xin K_infty$ there exists an $n$ big enough so that the connected component of $x$ in $K_n$ is a subset of $mathcal U$), the map $Xrightarrow X’$ is an homeomorphism.

Now, one could try to define a " pro-cellular" homology for such a projective limit of CW-complexes : $X^{(0)}$ would be the 0-dimensional subset of $X$ given by all the vertices of the CW-complexes plus $K_{infty}$, then $X^{(1)}$ is given by all the edges of the CW-complexes,…

This cellular decomposition fails to be a CW-complex in several ways but the relative homology defining the cellular chain complex $H(X^{(n)},X^{n-1};G)$ is still composed of free $G$-modules and is what one would expect for a CW-complex i.e. the free module generated by the $n$-facets.

Does one still have equivalence with singular homology?, I guess the answer is No but I don’t see why.

MathOverflow Asked by Léo Brunswic on January 27, 2021

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