= CW filtration
{c}
The <CW filtration> of a <CW complex> is its skeletal sequence
$$
\varnothing=X^{-1}\subseteq X^0\subseteq X^1\subseteq\cdots\subseteq X.
$$
Here $X^0$ is the discrete <set> of <vertices>, and $X^n$ contains all <topological cells> of <dimension> at most $n$. For $n\geq1$, collapsing $X^{n-1}$ turns the boundary of every attached $n$-<disc> into the same <base point>, giving
$$
X^n/X^{n-1}\cong\bigvee_{\text{$n$-cells}}S^n.
$$
In 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>.
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