... So the topology on $V$ is determined by the neighborhoods of zero. Most topologies that we shall need are defined by a directed set of continuous semi-norms $(p_i)_{i \in I}$. A subset $U \subseteq V$ is a neighborhood of $x \in V$ if and only if there are $i \in I$ and $\epsilon > 0$ such that all $v \in V$ with $p_i(v-x) < \epsilon$ belong to $U$.
Exercise: Let $V$ be a vector space and let $(p_i)_{i \in I}$ be a directed set of semi-norms on $V$. Show that the topology defined above makes $V$ a topological vector space.
This is a very simple exercise in some lecture notes (not homework). But I am not familiar with the terms such as topology, topological vector space, and directed set, though I studied some elementary functional analysis. I am trying to discuss the proof below but many mistakes may be there. Please review and point out my mistakes and, perhaps, write some comments or a detailed answer. Thank you in advance!
To show $(p_i)_{i \in I}$ defines a topology on $V$, by definition, I think $(p_i)_{i \in I}$ needs to define a family of open subsets of $V$? But $(p_i)_{i \in I}$ as a directed set of semi-norms that are non-negative real numbers, how could they become the subsets of a vector space $V$? Let $(p_i)_{i \in I}$ induces a directed set of pseudo-metric such that $d_i(x,y) = p_i(x-y)$, then we may define a net of neighborhoods of any $x \in V$ on the directed set, say $B_{\epsilon_i}(x) = \{y \in V|d_i(x,y)<\epsilon_i\}$, which are subsets of $V$. Right? Then I need to check if the neighborhoods defined above are open, i.e., if the net $\{B_{\epsilon_i}(x)\}$ satisfy the properties of a topology. But how? Obviously, the empty set $\emptyset \in \{B_{\epsilon_i}(x)\}$ as $\epsilon_i = 0$. The whole space $V \in \{B_{\epsilon_i}(x)\}$ as $\epsilon_i \to \infty$? The infinite union of the neighborhoods $\bigcup_{i}^\ B_{\epsilon_i}(x) \subseteq \{B_{\epsilon_i}(x)\}$ and the finite intersection of the neighborhoods $\bigcap_{i} B_{\epsilon_i}(x) \subseteq \{B_{\epsilon_i}(x)\}$? Why? I think this proof is not correct since I am still discussing about a metric topology but the question asks for a general topology. And I don't understand why the directed set of$(p_i)_{i \in I}$, rather than just a set, is necessary.