 CW filtration

The CW filtration of a CW complex is its skeletal sequenceHere $X^0$ is the discrete set of vertices, and $X^n$ contains all topological cells of dimension at most $n$. For $n\ge 1$, collapsing $X^{n-1}$ turns the boundary of every attached $n$-disc into the same base point, givingIn degree zero, a pointed cofiber convention uses $X_{+}^0$, with an extra disjoint base point; it is a wedge sum of one $S^0$ for every original vertex. Without adding that extra base point, the initial object is just the discrete vertex set. This distinction fixes the zero-degree convention in cellular homology.