Deck transformations and compact CW complexes Let $X$ be a CW complex and $\widetilde{X}$ its universal cover,  formed by lifting the CW structure on $X$. A finite cellular cochain, denoted $\phi$, is a cochain in $H^n(\widetilde{X}^n,\widetilde{X}^{n-1};\mathbb{Z})$ for any $n$ (note we are using cellular cohomology) such that $\phi$ maps all but finitely many $n$-cells to zero. 
In the algebraic topology book I am reading, it is stated (without proof) that if $X$ is a finite CW complex, then finite cellular cochains are exactly homomorphisms $\psi:H_n(\widetilde{X}^n,\widetilde{X}^{n-1}) \to \mathbb{Z}$ such that for every $n$-cell $e^n$ of $\widetilde{X}$, $\psi(\gamma e^n)$ is nonzero for only finitely many covering transformations $\gamma \in \pi_1(X)$. Here we are using the implicit bijection between $\pi_1(X)$ and covering transformations of $\widetilde{X}$. 
My question is this: doesn't this result also hold if $X$ is a non-finite CW complex? The bijection between $\pi_1(X)$ and the group of covering transformations still works if $X$ is infinite, so why doesn't the result?
Thank you in advance.
 A: If $X$ has infinitely many $n$-cells, then a homomorphism $\psi:H_n(\widetilde{X}^n,\widetilde{X}^{n-1})\to\mathbb{Z}$ might be "finite" on each orbit of $\pi_1(X)$, but still be "infinite" overall because there are infinitely many orbits of $\pi_1(X)$ among the $n$-simplices of $\widetilde{X}$.  For instance, if $X$ is any simply connected CW-complex, then $\widetilde{X}=X$ and the condition that for each $e^n$, $\psi(\gamma e^n)$ is nonzero for only finitely many $\gamma$ is trivially satisfied by any $\psi$ (since there is only one choice of $\gamma$!).  But $X$ might have infinitely many $n$-cells, in which case not every $\psi$ is finite.
In general, there is an action of $\pi_1(X)$ on the set $\widetilde{S}$ of $n$-cells in $\widetilde{X}$, and the orbits of this action are in bijection with the set $S$ of $n$-cells in $X$.  A map $\phi:\widetilde{S}\to\mathbb{Z}$ defines a finite cochain iff it has finite support, and it satisfies your second condition iff its restriction to each orbit of $\pi_1(X)$ has finite support.  When $X$ is a finite CW-complex, $S$ is a finite set, so there are only finitely many orbits, so if $\phi$ has finite support on each orbit, it has finite support on all of $\widetilde{S}$ (a finite union of finite sets is finite).  But if there are infinitely many orbits, just knowing the support is finite on each orbit doesn't tell you the support is finite on all of $\widetilde{S}$.
