Let $\alpha$ be an ordinal, considered as a set of the ordinals less than it. Aditionally suppose we have a function $h:\alpha+1 \to \mathbb N$. Suppose for each $i \leq \alpha$ we have a topological space $X_i$, and further that:
- If $i$ is a successor ordinal, then $X_i$ is obtained by attaching any number (cardinality) of $h(i)$-cells to $X_{i-1}$. I.e. $X_i$ is the pushout $\coprod_k D^{h(i)} \leftarrow \coprod_k S^{h(i)-1} \to X_{i-1}$, with $k$ being any cardinal.
- If $i$ is a limit ordinal, then $X_i$ is the colimit of $X_{j < i}$. I.e. $X_i=\bigcup_{j < i} X_j$ with the weak topology.
Under this assumptions, we say that $X_\alpha$ is a cell complex over $X_0$ of height $\alpha$. (Here $X_0$ can be any space).
My question is: is any cell complex over $X_0$ of height $\alpha$ homeomorphic to a cell complex over $X_0$ of height $\omega$?
This is motivated by the fact that the references I've read don't specify if $\alpha$ must be $\omega$, Other places mention transfiniteness though don't mention the limit case, like this MO post. My hope is that maybe the chosen $\alpha$ doesn't matter?
Let $X$ be a transfinite cell complex, and define a relation $\prec$ on the set $C$ of (open) cells of $X$ where $a\prec b$ if the image of the attaching map for $b$ intersects $a$. For any cell $b$, there are only finitely many $a$ such that $a\prec b$, since the image of the attaching map for $b$ is compact (and by the usual argument, a compact subset of a transfinite cell complex can only intersect finitely many open cells). Also, $\prec$ is a well-founded relation, since $a\prec b$ means that $a$ is attached in an earlier stage of the transfinite construction than $b$ is.
Now consider the rank function $r:C\to Ord$ of this well-founded relation $\prec$, i.e. $r(b)=\sup\{r(a)+1:a\prec b\}$. Since for any $b$ there are only finitely many $a$ such that $a\prec b$, an easy induction shows that in fact $r(b)$ is finite for all $b$. Now note that we could instead construct $X$ by starting with $X_0$ and attaching all of the cells of rank $0$, then all the cells of rank $1$, and so on. This constructs $X$ as a cell complex over $X_0$ of height $\omega$.