Weak topology on $\ell^2$ is an $\aleph_0$-space

[Moniker1998]

As the title.

In one of the articles $\aleph_0$-spaces on pi-base reference to https://topology.pi-base.org/properties/P000179/references that is $\aleph_0$-spaces by E. Michael, there's also contained a result that dual of a separable Banach space in its weak* topology is an $\aleph_0$-space, which shows $\ell^2$ in its weak topology is an $\aleph_0$-space.

#751

[Almanzoris]

Good result. I have just seen these PRs right now, it would have solved a few doubts I had about this space. Funnily enough, I have solved them earlier today, but deducing that it has a weaker property (Has a countable network), which would be your previous PR.

[Moniker1998]

Yes. My primary goal was to obtain separation properties for this space, but then I've seen one could just prove it has countable network, i.e. it's a cosmic space (which follows from $\ell^2$ having countable network), and here we have an even stronger property that it has countable k-network.