The following is an Example 6.7.8 in Spanier AT
If $C^*$ is the relative Alexander presheaf of $(X,A)$ (with some coefficient module $G$), the kernel of $\alpha:C^*\to\hat{C}^*$ is $C^*_0$ (the locally zero functions). To show that $\alpha$ satisfies condition (b) (which is gluing property), let $\varphi'\in \hat{C}^q(V,V\cap A)$ and assume $\varphi'$ represented by a compatible $\mathcal{U}$ family $\{\varphi_U\}_{U\in\mathcal{U}}$ where $\mathcal{U}$ is an open covering of $V$. Then $\varphi_U:U^{q+1}\to G$ for $U\in\mathcal{U}$ is locally zero on $U\cap A$ and $\varphi_U\mid (U\cap U')^{q+1} = \varphi_{U'}\mid (U\cap U')^{q+1}$ for $U,U'\in\mathcal{U}$. Therefore there is a function $\varphi:V^{q+1}\to G$ such that $\varphi\mid U^{q+1} =\varphi_U$ for $U\in\mathcal{U}$ and $\varphi(x_0,...,x_q) = 0$ if $x_0,...,x_q$ do not all lie in some element of $\mathcal{U}$. Then $\varphi$ is locally zero on $A$, whence $\varphi\in C^q(V,V\cap A)$ and $\alpha(\varphi) =\varphi'$.
Definition 1. Given a subset $A\subset X$, the relative Alexander presheaf of $(X,A)$ with coefficient $G$ denoted by $C^*(\cdot,\cdot\cap A;G)$ assigns to an open set $U\subset X$ the cochain complex $C^*(U,U\cap A;G)$. The description $C^*(U,U\cap A;G)$ is in Chapter 6.4.
Some terminologies are in Wikipedia.
Question is that in the example, it says '...and $\varphi(x_0,...,x_q) = 0$ if $x_0,...,x_q$ do not all lie in some element of $\mathcal{U}$.' and conclude $\varphi$ is locally zero on $A$. Why this is true? I think it should say $\varphi(x_0,...,x_q) =0$ if $x_i$'s are all in some element of $\mathcal{U}$ so that $\varphi$ is locally zero function. Could you explain this?