**Definitions :**



> 
> $(E,d)$ a metric space is finished-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 $i\in \mathbb N^\*$ ?
> 
> 
> 



About the finished, $\aleph_0$...-compactness
I think you are asking about *cardinal functions* on metric spaces. First, the property you call $\aleph\_0$-compact is more usually known as *Lindelöf*. As you say, it is well-known that for a metric space $E$, it is Lindelöf iff it is separable (iff it is second-countable iff it is c.c.c.). Of course, for topological spaces in general, no implication holds in either direction.


For metric spaces, this equivalence holds for the generalization of these properties to cardinals. A reference for this is the *Handbook of Set-Theoretic Topology*, edited by Kunen and Vaughan. 


The *density* (sometimes called *density character* to distinguish it from other meanings of density) $d(X)$ of a topological space $X$ is the minimal cardinality of a dense subset (this exists by well-orderedness of cardinals). So "$X$ is separable" is the same as $d(X) = \aleph\_0$. The *Lindelöf number* $L(X)$ is the smallest infinite cardinal $\kappa$ such that for each open cover there is a subcover of cardinality $\leq \kappa$. So "$X$ is Lindelöf" is the same as $L(X) = \aleph\_0$. (These definitions come from pages 11-12 of the aforementioned book.)


Now, for a metric space $E$, $d(E) = L(E)$ (see Theorem 8.1 (c) on pages 32-33), which is the generalization that I think you are looking for. Unfortunately, the proof given there treats the notion of a "$\sigma$-discrete base" and its existence in any metric space as well known, so I'll provide a little more information.


In the Handbook, $\sigma$-discrete is defined in chapter 9 (on page 350) and what you need to do to get a $\sigma$-discrete base in a metric space is to take the open cover of $E$ by open balls of radius $2^{-n}$ (for each $n \in \mathbb{N}$) and take a $\sigma$-discrete refinement of it using Theorem 2.8 of Chapter 9 (page 357). I suppose the reason the existence of a $\sigma$-discrete base is considered standard is that it is one direction of Bing's metrization theorem, and can be found in textbooks such as Engelking's *General Topology*, Theorem 4.4.3.


