Relative CW complex

Definition

Let $X$ be a topological space and let $A \subseteq X$ be a subspace. For $n \ge 0$ let $D^n$ be the closed unit ball in $\R^n$ with boundary $S^{n-1}$, where $S^{-1} = \emptyset$. A relative CW structure on $(X, A)$ is a sequence of subspaces $A = X^{-1} \subseteq X^0 \subseteq X^1 \subseteq \cdots$ of $X$ such that:

1. For each $n \ge 0$ there are a set $J_n$ and maps $\varphi_j \colon S^{n-1} \to X^{n-1}$, $j \in J_n$, with the following property. The space $X^n$ is the pushout of $X^{n-1} \leftarrow \bigsqcup_{j \in J_n} S^{n-1} \to \bigsqcup_{j \in J_n} D^n$, where the left map is given by the $\varphi_j$. That is, $X^n$ is the quotient of the disjoint union $X^{n-1} \sqcup \bigsqcup_{j} D^n$ that identifies each $x \in S^{n-1}$ in the $j$-th disk with $\varphi_j(x)$.
2. $X = \bigcup_{n \ge -1} X^n$.
3. A subset $C \subseteq X$ is closed if and only if $C \cap X^n$ is closed in $X^n$ for every $n$.

A relative CW complex is a pair $(X, A)$ with a relative CW structure. The images of the open disks $D^n \setminus S^{n-1}$ are the $n$-cells of $X$ relative to $A$.