Transitivity of CW-pairs

Question

If $(X,A)$ is a CW-pair with $A$ a subcomplex of $X$ and $(Y,X)$ is a CW-pair with $X$ a subcomplex of $Y$ is $(Y,A)$ a CW-pair with $A$ a subcomplex of $Y$?

Answer 1

A CW-complex is a topological space $Z$ plus a specific collection $\mathfrak E(Z)$ of subspaces (called cells) such that suitable conditions are satisfied. The set $\mathfrak E(Z)$ is called the CW-structure on $Z$. For the sake of precision let us write CW-complexes in the form $(Z,\mathfrak E(Z))$.

For $M \subset Z$ let us define $\mathfrak E(Z,M) = \{ e \in \mathfrak E(Z) \mid e \subset M \}$.

A subcomplex of $(Z,\mathfrak E(Z))$ is a CW-complex $(C,\mathfrak E(C))$ such that $C$ is a closed subspace of $Z$ and $\mathfrak E(C) \subset \mathfrak E(Z)$. In that case we have $\mathfrak E(C) = \mathfrak E(Z,C)$.

Aa CW-pair is a pair $((Z,\mathfrak E(Z)),(C,\mathfrak E(C)))$ of CW-complexes such that $(C,\mathfrak E(C))$ is a subcomplex of $(Z,\mathfrak E(Z))$. Alternatively we can define it as a tripel $(Z,C,\mathfrak E(Z))$ consisting of a topological space $Z$, a closed subspace $C \subset Z$ and a CW-structure $\mathfrak E(Z)$ on $Z$ such that $\mathfrak E(Z,C)$ is a CW- structure on $C$. Thus $(C,\mathfrak E(Z,C))$ is a subcomplex of $(Z,\mathfrak E(Z))$.

Let $(X,A,\mathfrak E(X))$ and $(Y,X,\mathfrak E(Y))$ be CW-pairs. Then $A$ is closed in $Y$ because closedness of subspaces is transitive. Since we require that $(X,\mathfrak E(X))$ is a subcomplex of $(Y,\mathfrak E(Y))$, we have $\mathfrak E(X) = \mathfrak E(Y,X)$. Thus $\mathfrak E(X,A) \subset \mathfrak E(X) = \mathfrak E(Y,X) \subset \mathfrak E(Y)$.

Therefore $(A,\mathfrak E(X,A))$ is subcomplex of $(Y,\mathfrak E(Y))$.

However, you can also define a CW-pair $(Z,C)$ to be a pair consisting of a topological space $Z$ and a closed subspace $C \subset Z$ such that $Z$ admits some CW-structure $\mathfrak E(Z)$ such that $\mathfrak E(Z,C)$ is a CW-structure on $C$. Let us call any such $\mathfrak E(Z)$ an admissible CW-structure for $(Z,C)$. Note that with this definition no specific CW-structure is associated with $(Z,C)$.

Given CW-pairs $(X,A)$ and $(Y,X)$ in that sense, we cannot be sure that we find an admissible CW-structure $\mathfrak E(Y)$ for $(Y,X)$ such that the induced CW-structure $\mathfrak E(Y,X)$ on $X$ is at the same time an admissible CW-structure for $(X,A)$. Certainly $A$ is closed in $Y$, but that is all we can say.