topology of relative CW complex

Question

I understand the construction of relative CW complex. However, I don't get the topology of a relative CW complex. I am not sure what properties of CW complex transfer to a relative CW complex. For a normal CW complex $X$, a set $O \subset X$ is open iff $O \bigcap X_n$ open for each n. Does this also hold for relative CW complex?

Answer 1

Yes, it is correct. In the case of a relative CW-complex we start with a space $A = X^{-1}$, attach $0$-cells to $X^{-1}$ and get a space $X^0$ (i.e. $X^0$ is the topological sum of $A$ and a discrete space $D^0$), then attach $1$-cells to $X^0$ and get a space $X^1$ etc.

This construction yields an ascending sequence of topological spaces $X^n$. As for an ordinary CW-complex, $X = \bigcup_n X^n$ is given a topology by defining $O \subset X$ to be open if all $O \cap X^n$ are open in $X^n$.