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112 the case. |
112 the case. |
113 (Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) |
113 (Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) |
114 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
114 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
115 refinements with both $K$ and $K'$. |
115 refinements with both $K$ and $K'$. |
116 Let $KL$ and $K'L$ denote these two refinements. |
116 Let $KL$ and $K'L$ denote these two refinements. |
117 Then filtration degree 1 chains associated to the four anti-refinemnts |
117 Then filtration degree 1 chains associated to the four anti-refinements |
118 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
118 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
119 give the desired chain connecting $(a, K)$ and $(a, K')$ |
119 give the desired chain connecting $(a, K)$ and $(a, K')$ |
120 (see Figure \ref{zzz4}). |
120 (see Figure \ref{zzz4}). |
121 |
121 |
122 \begin{figure}[!ht] |
122 \begin{figure}[!ht] |