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104 |
104 |
105 Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. |
105 Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. |
106 We want to find 1-simplices which connect $K$ and $K'$. |
106 We want to find 1-simplices which connect $K$ and $K'$. |
107 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
107 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
108 the case. |
108 the case. |
109 (Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) |
109 (Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) \scott{Why the $x^2$ here?} |
110 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
110 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
111 refinements with both $K$ and $K'$. |
111 refinements with both $K$ and $K'$. |
112 Let $KL$ and $K'L$ denote these two refinements. |
112 Let $KL$ and $K'L$ denote these two refinements. |
113 Then 1-simplices associated to the four anti-refinements |
113 Then 1-simplices associated to the four anti-refinements |
114 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
114 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |