Definitions :

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

$(E,d)$ is $\aleph_0$-compact if for any infinite covering of $E$ by open, we can extract a countable subcover

Remark :

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

we known the space $(E,d)$ with $\aleph_0$-compactness is exactly the space $(E,d)$ separable.

Question :

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