Définitions :

$(E,d)$ a metric space is $\mathbb N$-compact if any recovery of $E$ by open, we can extract a finite recovery

$(E,d)$ is $\aleph_0$-compact if any infinity recovrey of $E$ by open, we can extract a denombrable recovery

Remark :

we can imagine what's $\aleph_i$-compacity.

we known the sapce $(E,d)$ with $\aleph_0$-compacity is exactely the space $(E,d)$ separable.

Question :

What is we knowning about the $\aleph_i$-compacity for $i\in \mathbb N^*$ ?