neighborhood of $X_j$, we define element $(f,a)\in A_+$ corresponding to $X_j$ and $W_j$ as follows:

For each $t\in[0,1]_i,\,i=\{1,2,\cdots,l\}$, define $$f(t,i)=
 \begin{cases}
  \varphi_0^i(f_{11}^j)\dfrac{\eta-dist(t,[0,a_i\eta]_i)}{\eta}, & \mbox{if } 0\leq t\leq (a_i+1)\eta \\
  0, & \mbox{if } (a_i+1)\eta\leq t\leq (b_i-1)\eta \\
  \varphi_1^i(f_{11}^j)\dfrac{\eta-dist(t,[b_i\eta,1]_i)}{\eta}, & \mbox{if } (b_i-1)\eta\leq t\leq 1
\end{cases}$$ All such elements $(f,a)=(f(t,1),f(t,2),\cdots,f(t,l))\in A_+$ are included in the set $H(\eta)$ and are called test functions of type 1.

(b)For each closed subset $X=\cup_s[x_{r_s},x_{r_{s+1}}]_i \subset [\eta,1-\eta]_i$(the finite union of closed intervals $[x_r,x_{r+1}]$ and points $\{x_r\}$). So there are finite subsets for each $i$. Define $(f,a)$ corresponding to $X$ by $a=0$ and for each $t\in(0,1)_r,r\neq i,f(t,r)=0$ and for $t\in(0,1)_i$ define $$f(t,i)=
\begin{cases}
  e_{11}^i\big(1-\dfrac{dist(t,X)}{\eta}\big )e_{11}^i, & \mbox{if } dist(t,X)<\eta \\
  0, & \mbox{if } dist(t,X)\geq \eta.
\end{cases}$$ here we use $e_{11}^i$ as the matrix units of $M_{l_i}(\mathbb{C})$. All such elements are called test functions of type 2.

Note that for any closed subset $Y\subset[\eta,1-\eta]$, there is a closed subset $X$ consisting of the union of the intervals $[x_r,x_{r+1}]$ such that $X\supset Y$ and for any $x\in X$, $dist(x,Y)\leq\eta$.

[ Let $\phi:A\rightarrow M_n(\mathbb{C})$ be a homomorphism. $\phi$ has the expression as 1.3. For a closed neighborhood $W_j$ of $X_j=\{\theta_j\}$, define $$\#_{X_j}\big (Sp\phi\cap W_j \big):=
 t_j+\sum_{i=1}^{l}\alpha_{ij}\#\big (Sp\phi\cap [0,a_i\eta]_i\big)+\sum_{i=1}^{l}\beta_{ij}\#\big (Sp\phi\cap 