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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?

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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.

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  • $\begingroup$ it's cute; I'm not sure where to comment this, answer or question --- but I think something here is avoiding that CW complexes are more about modeling homotopy types than homeomorphisms, and this particular example has the homotopy type of a countably-infinite bouquet. In fact, it's a theorem that any continuous map of CW complexes is homotopic to a filtered map in the sense of maping $n$-skeleta to within $n$-skeleta. $\endgroup$
    – some guy on the street
    Commented May 4, 2010 at 21:04
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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.

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  • $\begingroup$ Yes. You can see this also by cellular approximation and the Whitehead theorem. $\endgroup$
    – Florian
    Commented May 4, 2010 at 19:12
  • $\begingroup$ I'm not exactly sure what you mean, but it sort of sounds like you're trying to apply the Whitehead theorem to something in order to conclude it's homotopy equivalent to a CW complex, and the Whitehead Theorem requires that you start with spaces of the homotopy type of CW complexes. I could be wrong, but I don't think you can get away without doing at least some real work here (as in Milnor's book). $\endgroup$
    – Dan Ramras
    Commented May 4, 2010 at 19:19
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    $\begingroup$ These objects are cofibrant (and of course fibrant) in the usual model structure on topological spaces. Therefore every weak homotopy equivalence between them is a homotopy equivalence. But it is "real work" to show that this is in fact a model category. $\endgroup$
    – Florian
    Commented May 4, 2010 at 19:25
  • $\begingroup$ Good point! So any CW approximation to such a space is automatically a homotopy equivalence, e.g. the map from the singular complex is an equivalence. That's very nice. (Maybe this is what was meant by "cellular approximation" in the first comment; I usually think of cellular approximation as meaning that any map between CW complexes is homotpic to a cellular map, hence my confusion.) $\endgroup$
    – Dan Ramras
    Commented May 5, 2010 at 18:27
  • $\begingroup$ Let me also point out that there are two model structures on topological spaces, one with ordinary weak equivalences as the model categorical weak equivalences (due to Quillen) and the other with honest homotopy equivalences as the weak equivalences (due to Strom). We're talking about the former model structure here. $\endgroup$
    – Dan Ramras
    Commented May 5, 2010 at 18:29

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