Is there an uncountable subspace $F \subseteq \omega^{\omega_1}$ which is closed and discrete?

Question

Does there exist an uncountable subspace $F \subseteq \omega^{\omega_1}$ which is closed and discrete?

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Edit: I made a couple attempts to solve it, but they didn't get anywhere (also I don't believe they're interesting, but just in case...).

1. The classic argument that the square of the Sorgenfrey line, $X^2$, is not T4 uses the fact that it is separable and has a discrete closed subspace of size $2^{\aleph_0}$. Namely, if it were T4, then by the Tietze extension theorem there would be at least $2^{2^{\aleph_0}}$ continuous functions $X^2 \to \mathbb{R}$. On the other hand, from separability the number of such functions is at most $2^{\aleph_0}$, which is a contradiction, since $2^{\aleph_0} < 2^{2^{\aleph_0}}$.

   But the analogous argument doesn't seem to work here, firstly because I don't know if $\omega^{\omega_1}$ is normal* (I do know it is separable), secondly because we only get that

   $2^{\aleph_1} \leqslant |C(\omega^{\omega_1})| \leqslant 2^{\aleph_0}$

   and that is consistent with ZFC.

2. I don't know whether $\omega^{\omega_1}$ is a Lindelöf space*. If it is, it follows that the answer is negative: suppose $F \subseteq \omega^{\omega_1}$ is a closed discrete subspace. Then as a closed subspace of a Lindelöf space, it is also Lindelöf, but since it's discrete, it must be countable.

3. Also since $\displaystyle \omega = \bigcup_{n < \omega} n$ and $n^{\omega_1}$ is compact, I feel like $\omega^{\omega_1}$ should be close to being compact in a sense I am not able to formalize, but if it is something similar to being Lindelöf, maybe it would be of use here.

The question was conceived by myself, so I have no idea how difficult it is or whether the answer is decidable in ZFC.

*I would also be interested in answers to these two questions (if $\omega^{\omega_1}$ is normal and if it is Lindelöf), but I don't know if I should ask them all at once.

Answer 1

Yes, there is such a subspace, according to this paper:

Mycielski J., $\alpha$-incompactness of $N^\alpha$, Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys. 12 (1964), 437–438.

(I cannot access this reference online, but it proves that $e(\omega^\tau) = \tau$ when $\tau$ is a cardinal smaller than the first weakly inaccessible cardinal. $e(X)$ is the supremum of cardinalities of a closed discrete subspace of $X$. So for $\tau=\omega_1$ it says that $\omega^{\omega_1}$ has a closed discrete subspace of size $\aleph_1$; if you can get hold of a digital copy, please send me one too, as I'm curious about its proof.)

I found the reference and result referred to in this paper which shows that $\omega^{\omega_1}$ is not $\alpha$-normal (a weaker property than normality defined in the referenced paper). This generalises the classical result due to A.H. Stone that $\omega^{\omega_1}$ is not normal (so also not Lindelöf, as a regular Lindelöf space would be normal), which is reproved on this blog

The cardinality argument you refer to is called Jones' lemma:

If $X$ is normal and has a dense subset of size $d(X)$ and a closed and discrete space of size $e(X)$, then $2^{e(X)} \le 2^{d(X)}$

and as $\omega^{\omega_1}$ is separable we know that if $\omega^{\omega_1}$ were normal, $2^{e(X)} \le \mathfrak{c} = 2^{\aleph_0}$, but this does not imply that $e(X) \le \aleph_0$, e.g $2^{\aleph_1} = 2^{\aleph_1}$ is possible, as you also state.

But non-Lindelöfness is clear both from non-normality of $\omega^{\omega_1}$ and Mycielski's theorem.