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521 \] |
521 \] |
522 \item |
522 \item |
523 Product morphisms are compatible with gluing (composition). |
523 Product morphisms are compatible with gluing (composition). |
524 Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
524 Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
525 be pinched products with $E = E_1\cup E_2$. |
525 be pinched products with $E = E_1\cup E_2$. |
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526 (See Figure \ref{pinched_prod_unions}.) |
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527 Note that $X_1$ and $X_2$ can be identified with subsets of $X$, |
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528 but $X_1 \cap X_2$ might not be codimension 1, and indeed we might have $X_1 = X_2 = X$. |
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529 We assume that there is a decomposition of $X$ into balls which is compatible with |
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530 $X_1$ and $X_2$. |
526 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
531 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
527 Then |
532 Then |
528 \[ |
533 \[ |
529 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
534 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
530 \] |
535 \] |