In this lecture we define the homology $H_{ \bullet}(X, \mathcal{F})$ associated to a topological space $X$ with a cell-like filtration $\mathcal{F}$ and prove that this homology is the same as the singular homology of $X$.

In the special case where $X$ is a cell complex and $\mathcal{F}$ arises from the skeleton filtration of $X$, we call this the cellular homology of $X$.

We conclude by giving an explicit formula for the boundary operator for a cell complex. This finally allows us to compute the homology of the real projective spaces, something that has eluded us so far.