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.

• 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:)

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I don't know what you mean by "satisfying the network". Anyway, there is a nice book called Counterexamples In Topology. It might interest you. – Gerry Myerson Jun 22 '12 at 1:30
It would probably useful to add the definition of network weight to your post, too. You probably meant $nw(X)\le\omega$ instead of $nw(X)<\omega$, since network weight cannot be finite by definition. – Martin Sleziak Jun 22 '12 at 11:20

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, $|X|\leq |P(\mathcal N)|=2^\omega$.