If $X\_1\supseteq X\_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H\_\*(\cap X\_i)$ to the inverse limit $\lim \, H\_\*(X\_i)$ is surjective. In particular, if there exist nonzero $\beta\_i\in H\_q(X\_i)$ such that the inclusion-induced homomorphism $i\_\*$ takes $\beta\_i$ to $\beta\_{i-1}$, is there an $\alpha\in H\_q(\cap X\_i)$ such that the inclusion-induced image of $\alpha$ is the $\beta\_i$ for each $i$? 


I have found Milnors paper "On the Steenrod homology theory" where he shows, using Steenrod homology theory, that there exists a surjective map $H\_q(\cap X\_i)\to \lim H\_q(X\_i)$. However, it is not completely clear from the text whether the preimage of $(\beta\_1,\beta\_2,\ldots)\in \lim H\_q$ is mapped to $\beta\_i$ by the inclusion-induced homomorphism $H\_q(\cap X)\to H\_q(X\_i)$.



Homology of infinite intersection
To expand on my answer above, consider the sequence $\dots\subset Y\_2\subset Y\_1$ 
of spaces where $Y\_i=X\_i$ for $i\leq n$ and $Y\_i=X\_n$ for $i>n$.


Milnors proof proceeds by taking $M\_i$ to be the mapping cylinder of $X\_{i+1}\rightarrow X\_i$ and $M\_0$ the cone over $X\_0$ with vertex $t$. Then one takes the union $F=\cup M\_i$ with $M\_i$ glued to $M\_{i+1}$ along $X\_{i+1}$. Then one makes a space $T=F\cup X$ by adding $X$ `at infinity'. The upshot is that $T$ can be contracted onto $t$ by collapsing everything down to $X\_1$ and then using the cone $M\_0$ to get down to $t$.


Now, once this is done, Milnor sets $F\_1$ to be the disjoint union of the odd $M\_i$
and $F\_0$ the disjoint union of the even $M\_i$. Then $F\_0\cap F\_1$ is just the disjoint union of all the $X\_i$, and $F\_0\cup F\_1 = F$.


The proof now follows quickly by looking at the long exact sequence i homology corresponding to the triple $(T,X\cup t,t)$ together with Mayer-Vietoris for $(F\_0,F\_1)$.


Now carry out the same constructions for $Y$, denoting the relevant spaces by $M\_i', t', T', F',F\_0',F\_1'$.


The point is that there are natural maps $M\_i'\rightarrow M\_i$ inducing maps $T'\rightarrow T$ and $F'\rightarrow F$, and the induced maps on cohomology are evidently the ones gotten by inclusion. 


Now, $Y=\cap\_m Y\_m = X\_n$ and so the map Milne gets $H\_q(X)\rightarrow \lim H\_q(X\_i)$ commutes with the inclusion maps $H\_q(X)\rightarrow H\_q(X\_n)$, which gives what you want.


