I am currently reading through May's "Algebraic Topology" and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using the canonical homeomorphism $(D^{n}, S^{n-1}) \simeq (D^{p} \times D^{q}, D^{p} \times S^{q-1} \cup S^{p-1} \times D^{q})$.

However, I remember hearing that a product of CW-complexes need not be again a CW-complex, ie. the topology on it is not the correct one. Here, May is working in the category of compactly generated spaces and the topology on this product can in general be finer than the usual product topology.

Is it easy to see that is this case the topology on the product of CW-complexes is the right one?

  • 2
    $\begingroup$ Well, I don't know whether the fact you asked is easy or not (to you), but there is a reference for you to go over : www.math.cornell.edu/~hatcher/AT/AT.pdf (Hatcher, Algebraic topology), Theorem A.6. $\endgroup$ – cjackal Dec 1 '12 at 16:08
  • $\begingroup$ I agree that the reference touches the subject I mention, but it gives sufficient conditions for the product in category of all topological spaces to be "the right one". My question is whether the compactly generated topology is always the right one. $\endgroup$ – Matthew Dec 1 '12 at 17:56
  • $\begingroup$ Theorem A.6 says that if X,Y are CW complex (so objects in the category of compactly generated spaces) then the product of X & Y in this category also has the CW structure. I think this is exactly what you want. $\endgroup$ – cjackal Dec 1 '12 at 20:13

Let $X$ and $Y$ be CW complexes. Let $X\times Y$ be the usual product (in $\mathbf{Top}$, the category of all spaces), and let $X\times_k Y$ be the $k$-ification of $X\times Y$, that is the topology coherent with the compact subsets of that space, so $X\times_k Y$ is the product in the category of $k$-spaces. By $X\times_{CW} Y$ we will denote the CW structure on the product.

Since every CW complex is a $k$-space, being a quotient of a topological sum of balls, which are compact Hausdorff spaces, so is $X \times_{CW} Y$. The projections from $X \times_{CW} Y$ to $X$ and to $Y$ induce the identity map to $X\times Y$, and thus a continuous identity map $i: X \times_{CW} Y \to X\times_k Y$, by universality of the $k$-ification.

In order to show that the inverse $j: X \times_k Y \to X \times_{CW} Y$ is continuous, we show that $j|_K$ is continuous on every compact subset $K$ of $X\times Y$. Since the both projections of $K$ are compact, they are contained in finite subcomplexes $C_X \subset X$ and $C_Y\subset Y$. Their product $C_X\times C_Y$ is compact Hausdorff and thus a $k$-space, and it is a subspace of $X\times_k Y$ containing $K$. Since $C_X \times_{CW} C_Y \to C_X \times C_Y$ is a continuous map from a compact to a Hausdorff space, it is a homeomorphism. That means $j|_K$ embeds $K$ into $C_X\times_{CW} C_Y$ and thus into $X \times_{CW} Y$.

I hope everything is correct, it is easy to lose track with all these different topologies flying around, so I urge you to double check what I did here.


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