Let $K$ be a field with valuation $v:K\to G\cup\{\infty\}$ where $G$ is an ordered abelian group. In section 7.62 of the book "Foundations of analysis over surreal number fields." Vol. 141. Elsevier, 1987, by Alling, Norman L., the valuation topology on $K$ is defined as the one with base $\{B(x_0,>g):g\in G,x_0\in K\}$ where $$B(x_0,>g)=\{x\in K:v(x-x_0)>g\}.$$
He shows that $K$ equipped with this topology is a Hausdorff topological field, and if $K$ is an ordered field then the order topology coincides with the valuation topology.
Until here everything seems normal, but then in another section (7.64) he defines the modified valuation topology on $K$ as the one with base $\{B(x_0,\geq g):g\in G,x_0\in K\}$ where $$B(x_0,\geq g)=\{x\in K:v(x-x_0)\geq g\}.$$
He shows that $K$ equipped with this topology is a Hausdorff topological field, and if $K$ is an ordered field then the order topology coincides with the modified valuation topology.
If I am not mistaken, the valuation topology and the modified valuation topology are always identical (when $G$ is not trivial, independently if $K$ is ordered or not) because of the inclusions: $$B(x_0,\geq 2|g|)\subset B(x_0,> g)\subset B(x_0,\geq g),$$ where $|g|=\max\{g,-g\}$.
Questions: Am I mistaken? Do these topologies always coincide? Why is Norman L. Alling treating these topologies as different ones? Am I missing something?
