Another similar notion to compactness

[mad mathematician]

I really don't understand why I didn't think about it while I learned UG topology back then.

The usual definition of Compactness is that every open cover of the compact space can be covered by a finite union from the arbitrary cover.

What about looking at a set that is covered by the intersection of a finite number of closed sets, from which we can cover by any number of closed sets' interesection?

I.e, if $X=\cap_{i=1}^n F_i$ then one can cover $X$ by an intersection of any number of closed sets.

Is there such a notion? is it too trivial? too strong?

 

[wrobel]

Any set can be covered by a finite intersection of closed sets; simply take its closure. Furthermore, the closure of a set is itself the intersection of all closed sets containing it.

 

[martinbn]

It is hard to say what you are thinking of, but this might be interesting to you
https://en.wikipedia.org/wiki/Finite_intersection_property#Compactness