In Spanier's book Algebraic Topology chapter 6, Theorem 16, as the following picture, it says
where a locally trivial presheaf $\Gamma $ means ${\Gamma _x} = 0$ for all $x$.
How can I take such open neighborhood $V_x$ if $x$ is in the boundary of some ${U_{{\alpha _0}}} \cap \cdots \cap {U_{{\alpha _q}}}$?
If $x$ is contained in ${U_{{\alpha _0}}} \cap \cdots \cap {U_{{\alpha _q}}}$, then we can choose an open neighborhood $U_x$ of $x$ such that ${\left. {\varphi ({U_{{\alpha _0}}}, \cdots ,{U_{{\alpha _q}}})} \right|_{{U_{{\alpha _0}}} \cap \cdots \cap {U_{{\alpha _q}}} \cap {U_x}}} = 0$.
However, if $x$ is contained in the boundary of ${U_{{\alpha _0}}} \cap \cdots \cap {U_{{\alpha _q}}}$, then ${U_{{\alpha _0}}} \cap \cdots \cap {U_{{\alpha _q}}} \cap {U_x} \ne \phi $ for all open neighborhoods of $x$, and I just don't know how to take a neighborhood of $x$ small enough such that we can have ${\left. {\varphi ({U_{{\alpha _0}}}, \cdots ,{U_{{\alpha _q}}})} \right|_{{U_{{\alpha _0}}} \cap \cdots \cap {U_{{\alpha _q}}} \cap {U_x}}} = 0$