4

There is a standard CW-topology on the finite product $X\times Y$ of CW-complexes $X$ and $Y$. Is there a standard CW-topology on an infinte prodcut $\prod_{n=1}^{\infty}X_{n}$ of CW-complexes? With the usual product topology such spaces are not even locally contractible. If so, what would this structure be explicitly for an infinte product of spheres $\prod_{n=1}^{\infty}S^k$?

Edit: Perhaps this question is more explicit. If $CW$ is the category of CW-complexes does the forgetful functor $CW\to Set $ preserve (at least countably infinite) products? A positive answer would imply that you can identify a CW-structure on $\prod_{n=1}^{\infty}X_{n}$ which would provide a topology finer than the usual product topology.

J.K.T.
  • 1,620
  • 1
    When you say CW-topology, do you mean CW-structure? – Dan Rust Nov 20 '13 at 17:49
  • Yes, I guess so. The CW-topology is just the final topology with respect to the attaching maps so I guess if it has a CW-structure then the topolgy is already determined. – J.K.T. Nov 20 '13 at 18:00
  • Is there a way to attach k-cells to $\bigvee_{n=1}^{\infty}S^{k-1}$ to obtain a CW-complex whose underlying set is $\prod_{n=1}^{\infty}S^k$ and which behaves like a categorical product in the category of CW-complexes? This would answer my question about spheres. – J.K.T. Nov 20 '13 at 18:05
  • Well for $k=1$, $\bigvee_{n=1}^{\infty} S^0=\mathbb{Z}$ and attaching $1$-cells can only give us some 1-dimensional CW complex so can't be homotopy equivalent to the infinite torus $\prod_{n=1}^{\infty}S^1$. This fails in higher dimensions too for similar reasons. (But this doesn't seem to relate to the question you asked in your post which is more interesting). – Dan Rust Nov 20 '13 at 18:12
  • Perhaps my suggestion about attaching k-cells was off but the infinte torus is clearly not homotopy equivalent to any CW-complex. It is not even semilocally simply connected. The question is how to change the topology of the infinite torus (or determine a CW-structure) so it becomes a CW-complex. – J.K.T. Nov 20 '13 at 18:21
  • If you're changing the topology then you're losing any structure beyond the structure of a set, so it's not clear what you're asking. Of course any set of cardinality equal to the cardinality of the continuum has a compatible structure of a CW complex because you can just lift a topology up from the real line via a bijection. This isn't very useful. – Dan Rust Nov 20 '13 at 18:24
  • 1
    I find difficult to believe that an infinite product operation wouldn't be useful in algebraic topolgy. Clearly I don't want just any old topology (as indicated by the comparison to the finite case). I want to know if $\prod X_n$ can be given a CW-structure on a point-set level (not up to homotopy) which gives the categorical product of CW-complexes. If possible this would be equivalent to changing the topology in a natural way. – J.K.T. Nov 20 '13 at 18:48
  • 2
    What morphisms do you have in the category? – Zhen Lin Nov 20 '13 at 19:33
  • Surely the obvious choice is to take the full subcategory of Top, since the OP's given no indication of wanting to pass to the homotopy category. – Kevin Carlson Nov 20 '13 at 19:42
  • My hope is that this would work when morphisms are just continuous functions but if not, I'd be willing to negotiate. – J.K.T. Nov 20 '13 at 19:44
  • 1
    This isn't an answer to whether the full subcategory of CW complexes has infinite products, but I think the standard algebraic topologist's answer would be to take the compactly-generated refinement of the product topology, and settle for the category of compactly generated (perhaps weakly Hausdorff) spaces as seen in Steenrod's "Convenient Category" paper. – Kevin Carlson Nov 20 '13 at 19:51
  • 1
    This is a fair point. On the other hand, if each $X_n$ is finite, the direct product $\prod_{n=1}^{\infty} X_n$ is metrizable and is already compactly generated weakly Hausdorff. – J.K.T. Nov 20 '13 at 20:20
  • I would be extremely surprised if any natural CW complex structure existed on an infinite product, though I don't know how to prove a negative like that. In particular, as for your question of whether the forgetful functor $CW\to Sets$ preserves infinite products, I suspect that is true rather trivially because the category $CW$ has no products in which infinitely many of the factors have positive dimension. – Eric Wofsey Jun 13 '17 at 07:22

0 Answers0