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Ok, so intuitively it's clear that the intersection of two subcomplexes of a CW-complex should be a subcomplex as well, but reading the inductive definition of a CW-complex, nowhere does it say that a cell should be attached to a whole other cell, that is to say: it seems to imply that i could, for example, attach a 2-cell to a point in the middle of a 1-cell as if there were a 0-cell there. But then the intersection of the 1-cell and the 2-cell in question would be a point that isn't a 0-cell, and therefore not a subcomplex.

Am I missing something from the definition?

Santiago Bosch
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  • What is your definition of a CW complex? – Sheel Stueber Aug 07 '18 at 01:25
  • the one that appears in Hatcher's "Algebraic Topology", you start by a discrete 0-skeleton and then attach cells via a map from the border of an n-disk to the n-1-skeleton. The thing is, that doesn't make it clear if the edge can go to any part of the skeleton or if it must attach to a full cell. – Santiago Bosch Aug 07 '18 at 14:10

1 Answers1

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IMO Hatcher's treatment of cell complexes leaves out a lot of the technicalities (which are important, considering that CW complexes are technical objects by design). There are two important things to notice:

  1. A subcomplex is a closed subspace
  2. Cells are open by definition

So the 2-cell that you're talking to is actually not a subcomplex. To make it so, you need to include all its limit points (being all the points where it attaches to the 1-frame). But a subcomplex consists of entire cells, so the subcomplex must actually include any cells onto which the 2-cell is glued (in your example, you need to include the entire 1-cell). And it cascades all the way down like this until you get the 0-cells that "support" the subcomplex.

So the 1-cell - while not being a subcomplex in itself - would "generate" the subcomplex consisting of itself and its endpoints. Similarly, the 2-cell generates the subcomplex consisting of itself, the 1-cell, and the 1-cell's two endpoints. From here it's clear that the intersection is again a subcomplex.

Joseph Granata
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