Traits for Cohen's modified product (S207)

[felixpernegger]

Right now this space is the most sparsely populated one.

[prabau]

P28 is redundant

[prabau]

P114: I don't think the mathse reference is needed here. It's a basic result about cardinalities that $\aleph_1\cdot\aleph_1=\aleph_1$. But if you really want to keep it, that's fine.

[felixpernegger]

P114: I don't think the mathse reference is needed here. It's a basic result about cardinalities that ℵ 1 ⋅ ℵ 1 = ℵ 1 . But if you really want to keep it, that's fine.

Its fine either way for me

[prabau]

P51: the subspace $\omega_1\times\{\omega_1\}$ is homeomorphic to S35 (ordinal space omega_1), which is scattered. So one could use that for the second part (together with an implicit meta-property: scattered is hereditary).
What do you think?

[felixpernegger]

Actually, is the product of (two) scattered spaces scattered? If yes, P51 follows since each factor is scattered and our topology is finer than the product.

[yhx-12243]

Yes, we can choose point by coordinate.