Is there necessarily a CW structure on a space build out of cells without demanding them to be attached in "right" order?


More precisely, let $X$ be a topological space such that the map $\emptyset\to X$ factorizes as a transfinite composition of inclusions
$$
\emptyset\to\ldots\to X\_\beta\to X\_{\beta+1}\to\ldots\to X
$$
where every map $X\_\beta\to X\_{\beta+1}$ is a pushout
$$
\begin{array}{rcl}
 S^{n-1} &\to& X\_\beta\\\
 \downarrow && \downarrow\\\
 D^{n} &\to& X\_{\beta+1}
\end{array}
$$
for some $n\geq 0$. Is $X$ necessarily a CW complex?



CW structure on spaces obtained by attaching cells wildly
Fix an irrational number $\alpha$.


Let $X\_2 = [0,1]$ (built from attaching a 1-cell to two 0-cells) and, for each larger $n$, let $X\_n$ be built from $X\_{n-1}$ with a new a 1-cell by attaching the ends to $0$ and the fractional part of $n \alpha$. Take $X$ to be the union.


There are no embeddings from an open disc $D^n$ into $X$ for n greater than 1. If $X$ admitted a CW-complex structure, this would force it to be 1-dimensional. However, $X$ cannot be homeomorphic to a 1-dimensional CW-complex, for example because the set of points which have no neighborhood homeomorphic to $\mathbb{R}$ do not form a discrete subspace.


