Let $X$ be a topological space. Assume that $X$ admits a finite decomposition of the form $X=\bigsqcup\limits\_{i=1}^n V\_i$ where each $V\_i$ is homeomorphic (in the subspace topology of $X$) to an open cell of dimension $d\_i$. By convention, the open cell of dimension $0$ is a point. We are not assuming that the disjoint union is a stratification of $X$, i.e., we allow the closure of $V\_i$ to intersect cells $V\_j$'s of dimension larger or equal to $d\_i$.


**Q:** Does it necessarily follow that $H\_k(X,\mathbb{Z})$ is a finitely generated 
$\mathbb{Z}$-module?



Homology of a finite disjoint union of open cells
This can fail badly even if the closure of each cell only intersects lower-dimensional cells. For instance, take an open disk together with some subset of the boundary that has infinitely many connected components, and glue it to an open interval by identifying the boundary set with a homeomorphic subset of the interval. The resulting space will not have finitely generated $H\_1$ (in fact, I think its $H\_1$ is uncountable).


