As a recap, the Lindelof number, $L(X)$, is the smallest infinite cardinal $\mathcal{K}$ so that every open cover of $X$ has a subcover with cardinality less than or equal to $\mathcal{K}$. Also the network weight, $NW(X)$, is the minimum of the cardinalites of the networks of $X$.
So I need to show $L(X) \leq NW(X)$. I attempt to do so below.
Proof
Let $\mathscr{U}$ be an open cover of $X$ and let $\mathscr{N}$ be a network of $X$. Let $\mathscr{V}$ be a subcover of $\mathscr{U}$. For each $U \in \mathscr{V}$, let $N_U \in \mathscr{N}$ satisfy $N_U \subset U$. Now define
$$f: \mathscr{V} \to \mathscr{N} \hspace{0.3cm} \mathrm{via} \hspace{0.3cm} f(U) = N_U$$
We show $f$ is injective and then we'll be done. Suppose $U_1, U_2 \in \mathscr{U}$ with $f(U_1) = f(U_2)$. Then $N_{U_1} = N_{U_2}$. Now, at this point it's not obvious to me that $U_1 = U_2$.
Intuitively, it is completely clear to me that every $U$ has at least one $N$ inside of it (and could have many more). So the idea of injectivity is so painfully clear, yet I am struggling justifying the injection. Any advice?
We are given an open cover $\mathscr{U}$ of $X$ and must construct a subcover $\mathscr{V}$ that is small enough. So picking a $\mathscr{V}$ to start with is not the right approach here. The following should work instead.
Let $\mathscr{N}$ be a network of $X$ of smallest cardinality (that is, of size equal to the network weight of $X$). Given an open cover $\mathscr{U}$ of $X$, consider the subset $\mathscr{N}'$ of $\mathscr{N}$ consisting of all the $N\in\mathscr{N}$ that are contained in some $U\in\mathscr{U}$. Then for each $N\in\mathscr{N}'$ choose some $U_N\in\mathscr{U}$ containing it. The collection $\{U_N:N\in\mathscr{N}'\}$ is a subcollection of $\mathscr{U}$ of size at most $NW(X)$ and it is easy to check that it covers $X$. So we have constructed a subcover of $\mathscr{U}$ of size at most $NW(X)$ as required.