If $(X, E)$ is a compact CW complex with CW subcomplex $(Y, E')$, then $H_k(X,Y)$ is finitely generated for every $k \ge 0$.
I see that to show $H_k(X,Y)$ is finitely generated I could also show that $H_k(W_*(X,Y))$ is finitely generated since they're isomorphic. But I'm having trouble proving that either are finitely generated.
I can see that $W_k(X,Y)$ is free abelian with rank equal to the number of $k$-cells in $E-E'$, but I don't see how I could show that this is finite.
Anyone have any ideas?
Definitions :
$W_*(X)$ is the chain complex where each $W_k(X)=H_k(X^k, X^{k-1})$ and the $X^k$ are a sequence of subspaces of $X$, $\{X^j : X^j \subset X^{j+1}\}$.
$W_*(X,Y)$ is the chain complex where each $W_k(X,Y) = H_k(X_Y^k, X_Y^{k-1})$ and $X_Y^k = X^k \cup Y$
Any compact CW-complex has only finitely many cells. Indeed, suppose $X$ is a compact CW-complex. Choose one point in the interior of each cell, and let $A$ be the set of these points. Then since each closed cell intersects the interiors of only finitely many cells, the intersection of $A$ with any closed cell is finite and in particular closed, so $A$ is closed in $X$. Since $X$ is compact, so is $A$. But by the same reasoning, every subset of $A$ is closed as well. Thus $A$ has the discrete topology and is compact, so it must be finite. Since $A$ has one point for each cell of $X$, we conclude that $X$ has only finitely many cells.