Intersection of subcomplexes 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?
 A: 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:


*

*A subcomplex is a closed subspace

*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. 
