I'm reading Lee's "Introduction to topological manifolds". I have a confusion with the first condition (closure finiteness) of the definition of the CW complex, namely: C condition says that the closure of each cell must be contained in the union of finitely many cells; on the other hand, since CW complex is a cell complex (i.e. a Hausdorff space with its partition), all cells have empty intersections. So the question is how the closure of any cell may be contained in the union of other cells (scince closure of a cell contains its interior points and all cells are disjoint, it follows that no other cell may contain iterior poits of this cell)? Thank you for your answers!
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$(0,1] \cap {0} = \emptyset$ but ${0} \in \overline{(0,1]}$. – Apr 16 '17 at 13:54
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Let $e$ denote a cell. Then $\overline{e}$ is contained in a finite union of cells and $e$ itself is one of these cells (so not "other" cells as you write). The cells different form $e$ will contain $\overline{e}-e$. – drhab Apr 16 '17 at 13:59
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drhab, thank you, I didn't take into account that e itself is allowed – Artem Bolshov Apr 16 '17 at 14:00