117 We want to find 1-simplices which connect $K$ and $K'$. |
117 We want to find 1-simplices which connect $K$ and $K'$. |
118 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
118 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
119 the case. |
119 the case. |
120 (Consider the $x$-axis and the graph of $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.) |
120 (Consider the $x$-axis and the graph of $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.) |
121 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
121 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
122 refinements with both $K$ and $K'$. (For instance, in the example above, $L$ can be the graph of $y=x^2+1$.) |
122 refinements with both $K$ and $K'$. (For instance, in the example above, $L$ can be the graph of $y=x^2-1$.) |
123 This follows from Axiom \ref{axiom:vcones}, which in turn follows from the |
123 This follows from Axiom \ref{axiom:vcones}, which in turn follows from the |
124 splitting axiom for the system of fields $\cE$. |
124 splitting axiom for the system of fields $\cE$. |
125 Let $KL$ and $K'L$ denote these two refinements. |
125 Let $KL$ and $K'L$ denote these two refinements. |
126 Then 1-simplices associated to the four anti-refinements |
126 Then 1-simplices associated to the four anti-refinements |
127 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
127 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |