Another follow up question to clarify what I believe is the last bit I'm missing.
The original question is here, while the first follow up is in this link.
Suppose we have two families of seminorms $p_\alpha$ and $q_\beta$. If for every $\alpha$ there is $\beta$ and a constant $C_{\alpha,\beta}$ such that $p_\alpha≤ C_{\alpha,\beta}q_\beta$ (and symmetrically for every $\beta$ there is $\alpha$) then they define the same topology. If the families are directed it suffices to consider only sufficiently "large" $\alpha,\beta$ because other seminorms are dominated by them. This is why the ordering helps.
The reference pointed was Simon's and Reed's book, where there's the definition:
Definition A family $\left\{ p_\alpha \right\}_{\alpha \in A}$ of seminorms on a vector space $V$ is called directed iff for all $\alpha,\beta \in A$ there's a $\gamma \in A$ and a $C$ so tht $$ p_\alpha(x) + p_\beta(x) \leq Cp_\gamma(x) $$ for all $x \in V$. Equivalently, by induction, for all $\alpha_1,\ldots,\alpha_n \in A$ there's a $\gamma$ and $D$ so that $$ p_{\alpha_1}(x) + \ldots + p_{\alpha_n}(x) \leq D p_\gamma(x) $$ for all $x \in V$.
I can't relate this definition though to the fact that we can just consider indices sufficiently large.
Can you clarify? Thank you very much
