I believe there are some topology spaces which satisfying the network weight is less than $\omega$, and its cardinality is more than $2^\omega$ (not equal to $2^\omega$), even much larger.
- Network: a family $N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x \in X$ and any nbhd $U$ of $x$ there exists an $M \in N$ such that $x \in M \subset U$.
Here I want to look for some simple topology spaces which are familiar with us. However, a little complex topology space is also welcome!
Thanks for any help:)