How could I define this $\mathrm{nw}(X)$ by using only one sentence?

Question

A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any neighbourhood $U$ of $x$ there exists an $M \in \mathcal N$ such that $x\in M \subset U$. $$\mathrm{nw}(X)=\min \{|\mathcal N|: \mathcal N \text{ a network for }X\} +\omega.$$

How could I define this $\mathrm{nw}(X)$ by using only one sentence?

For example:

The Lindelof number $l(X)$ of a topological space $X$ is the smallest number $\kappa$ such that every open cover of $X$ has a subcover the cardinality of which is at most $\kappa$.

Thanks for your time.

Answer 1

There is no real advantage to using just one sentence; in fact, the definition is probably clearer if you use two. However, it is possible to state it in one:

The net weight of a topological space $X$ is the smallest infinite cardinality $\kappa$ such that $X$ has a a family $\mathscr{N}$ of subsets, not necessarily open, such that every non-empty open set in $X$ is the union of members of $\mathscr{N}$, and the cardinality of $\mathscr{N}$ is at most $\kappa$; such a family is called a network for $X$.