Topology generated by Minkowski functionals

Question

Let $X$ be a locally convex topological vector space and let $\gamma$ be a local base of convex sets. Associate to each $C \in \gamma$ the Minkowski functional $\mu_C(x)= \inf(\lbrace t > 0 \mid t^{-1}x \in C \rbrace)$. Let $\mathcal{F}= \lbrace \mu_C(\cdot-x) \mid C \in \gamma\, , \, x \in X\rbrace$ be the collection of continuous Minkowski functionals. I want to show that the topology $\tau(\mathcal{F})$ generated by $\mathcal{F}$ coincides with the given topology $\tau_X$ of $X$, but I am unsure about what the notation $ \mu_C(\cdot-x)$ means. Does it mean the Minkowski functional $\mu_C(y-x)$ where $y$ is any element that varies over $X$? And further is there any reasonable way to argue from continuity of $\mu_C$ to there being equality $\tau(\mathcal{F})=\tau_X$, like if I can show that I can recover any $C \in \gamma$ via $\mu_C$ for any such $C$?