161 \begin{example} \label{sin1x-example} |
161 \begin{example} \label{sin1x-example} |
162 Consider the four subsets of $\Real^3$, |
162 Consider the four subsets of $\Real^3$, |
163 \begin{align*} |
163 \begin{align*} |
164 A & = [0,1] \times [0,1] \times [0,1] \\ |
164 A & = [0,1] \times [0,1] \times [0,1] \\ |
165 B & = [0,1] \times [-1,0] \times [0,1] \\ |
165 B & = [0,1] \times [-1,0] \times [0,1] \\ |
166 C & = [-1,0] \times \setc{(y,z)}{z \sin(1/z) \leq y \leq 1, z \in [0,1]} \\ |
166 C & = [-1,0] \times \setc{(y,z)}{e^{-1/z^2} \sin(1/z) \leq y \leq 1, z \in [0,1]} \\ |
167 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z \sin(1/z), z \in [0,1]}. |
167 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq e^{-1/z^2} \sin(1/z), z \in [0,1]}. |
168 \end{align*} |
168 \end{align*} |
169 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. |
169 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. |
170 Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, |
170 Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, |
171 and $\{D\}$ is a valid configuration of blobs in $C \cup D$, |
171 and $\{D\}$ is a valid configuration of blobs in $C \cup D$, |
172 so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. |
172 so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. |