CW structure on spaces obtained by attaching cells wildly 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?
 A: The space you get through such a process of cell attachments is at least homotopy equivalent to a CW complex, although maybe not homeomorphic (as Tyler's example shows).
This comes up in Milnor's Morse Theory (see pp. 21-24), where you build a CW complex inductively by attaching cells according to the critical points of a Morse function on a manifold.  There's no reason that the indices have to be increasing, and this causes one to attach cells "in the wrong order," i.e. a cell of dimension n may be attached to existing cells of dimension n or greater.  In his book on Morse Theory, Milnor shows that up to homotopy the resulting space is a CW complex.  The basic point is that each attaching map can be deformed to the appropriate skeleton, and attaching a cell along two homotopic maps produces two homotopy equivalent spaces (this is a bit tricky, and Milnor attributes it to Hilton).  Finally, an argument with homotopy colimits allows one to treat the case of countably many cell attachments.
I suppose that this applies to Tyler's example, and shows that the space he builds is homotopy equivalent to an infinite wedge of circles.
A: 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.
