Must $X$ be lindelöf if it has countable network?

Question

A network is like a base, except that its members need not be open sets.

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 $x\in M \subset U$.

Suppose $X$ has a countable network. Must $X$ be Lindelöf?

score 2 · Accepted Answer · answered Jun 06 '13 at 02:11

2

Yes. Let $\mathscr{N}$ be a countable network for $X$, and let $\mathscr{U}$ be an open cover of $X$. For each $x\in X$ choose $N_ x\in \mathscr{N}$ such that $x\in N_x\subseteq U$ for some $U\in\mathscr{U}$. Let $\mathscr{N}_0=\{N_x:x\in X\}$; clearly $\mathscr{N}_0$ is countable and covers$X$. $\mathscr{N}_0$ refines $\mathscr{U}$, so for each $N\in\mathscr{N}_0$ we can choose a $U_ N\in\mathscr{U}$ such that $N\subseteq U_N$. Then $\{U_N:N\in\mathscr{N}_0\}$ is a countable subcover of $\mathscr{U}$.

answered Jun 06 '13 at 02:11

Brian M. Scott

616,228

So it is also a hereditarily lindelof? – Paul Jun 06 '13 at 02:16
@Paul: Yes, it is. – Brian M. Scott Jun 06 '13 at 02:42
@Paul And hereditarily separable as well. Spaces with a countable network are called "cosmic spaces". Of course they include all countable spaces (singletons always form a network) and all second countable spaces. – Henno Brandsma Jun 06 '13 at 04:19
@Henno: I’d forgotten that; Paul might be interested in the WP entry for them. (I do think that it’s one of the silliest names around, though. I suppose that it’s from COntinous image of Separable MetrIC space.) – Brian M. Scott Jun 06 '13 at 04:23
@BrianM.Scott Ah, is that the reason they're called that way! Funny. – Henno Brandsma Jun 06 '13 at 04:25
@Henno Brandsma: Thanks for the name of cosmic space. I have heard it, however I don't know its definition. – Paul Jun 06 '13 at 06:06

Must $X$ be lindelöf if it has countable network?

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