Let $X$ be a topological affine space over a complete base field $\mathbb S := \mathbb C$, $\mathbb R$ or $\mathbb Q_p$. Let $X^*$ be the dual space of continuous affine functionals equipped with the weak-$*$ topology.

At this level of generality, what are the necessary properties that the weak-$*$ topology must satisfy? e.g., locally convex, Hausdorff, etc.

Here's a trivial property. Unlike the linear setting, the affine dual space contains a one-dimensional subspace of constant functionals $c(x) := c$.