Proof about certain family of sets that is a basis for the open-compact topology of homomorphism from a topological group to the unit circle

Question

Let $G$ be a locally compact topological group and $\hat{G}$ the set of homomorphisms from $G$ to the unit circle $\mathbb{S}^1 = \{ z \in \mathbb{C} : |z| =1 \}$. For every compact set $K \subseteq G$, $\epsilon >0$ and $f \in \hat{G}$, define $$T(f,K, \epsilon) = \{ g \in \hat{G} : \forall x \in K ( |fx- gx| < \epsilon )\}.$$ I am trying to prove that these form a basis for a topology in $\hat{G}$, but, I am not so sure on how to express an intersection of two of these sets as a union of them, I could use help or hints.