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This is theorem 2C.5 p.182 from Hatcher's Algebraic Topology: Every CW complex X is homotopy equivalent to a simplicial complex, which can be chosen to be of the same dimension as X, finite if X is finite, and countable if X is countable.


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In fact, your construction works for any small category $\mathscr A$ for which there is a functor $$ \widehat{\mathscr A} \to \mathsf{Cat},\, F \mapsto {\mathscr A}\big/{F} $$ where ${\mathscr A}\big/{F}$ is just another notation for the comma category $(\mathbf{yon}_{\mathscr A}\downarrow F)$. (This notation is justified by viewing $\mathscr A$ as a ...


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Intuitively you can think about the quotient space of such a good CW pair $(X,Y)$, as collapsing all cells in $Y$ to a point. So if you collapse $Y$ you get all cells of $X$ which were not in $Y$ but glued in maybe a very different way than before. So the dimension (i.e. the highest dimension where there exists a cell) of the quotient space decreases at ...



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