Looking for some simple topology spaces such that $nw(X)\le\omega$ and $|X|>2^\omega$ 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.


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*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:)

If $X$ is $T_0$ and $nw(X)=\omega$ (network weight) then $|X|\leq 2^\omega$. Let $\mathcal N$ be a countable network. For each $x\in X$ consider $N_x=\{N\in\mathcal N: x\in N\}$. Since $X$ is $T_0$ it follows that $N_x\ne N_y$ for $x\ne y$. Thus, $|X|\leq |P(\mathcal N)|=2^\omega$.    
