Wimpy powerset function

Define the 'wimpy powerset function' $\mathcal{W} : \mathrm{Set} \rightarrow \mathrm{Set}$ by writing $$\mathcal{W}(B) = \{X \in \mathcal{P}(B) : |X| < |B|\}.$$

A few preliminary observations.

1. If $B$ is finite, then $|\mathcal{W}(B)| + 1 = |\mathcal{P}(B)|.$

2. If $B$ is countable (e.g. take $B=\mathbb{N}$), then $|\mathcal{W}(B)| = |B|.$

What else is known about $\mathcal{W}$? In particular:

• What can we say about $\mathcal{W}(\aleph_1)$ and $\mathcal{W}(\beth_1)$?
• Do there exist sets $B$ such that $|\mathcal{W}(B)| = |\mathcal{P}(B)|$?

We're assuming ZFC, right?

$|\mathcal W(\omega_1)|=\beth_1$.

$\beth_1\le|\mathcal W(\beth_1)|\le\beth_2$;
if $2^{\aleph_0}=\aleph_1$, then $|\mathcal W(\beth_1)|=\beth_1$, but
if $2^{\aleph_0}=\aleph_2$ and $2^{\aleph_1}=2^{\aleph_2}=\aleph_3$, then $|\mathcal W(\beth_1)|=\beth_2$.

$|\mathcal W(\beth_{\omega})|=|\mathcal P(\beth_{\omega})|=\beth_{\omega+1}$.