What does "any collection" mean in this sentence? What does the expression "any collection" mean in the following sentence?

Prove that the intersection of any collection of compact sets is compact.

 A: Any collection of compact subsets of $X$ refers to any subset $\mathcal C=\{K_i|i\in I\}$ (elements indexed by some set $I$) of the power set $\mathcal P(X)$, that means each $K_i$ is a compact subset of $X$. The word family is often used. The set $\mathcal C$ can have any cardinality. It is often clear from the context if $\mathcal C$ is allowed to be empty or not. 
In the example you gave, they ask you to prove that the intersection of any family of compact subsets is compact. Since the intersection $\bigcap \mathcal C$ is all of $X$ if $\mathcal C$ is empty, but $X$ itself is not compact in general, we should assume that $\mathcal C$ is meant to be non-empty in this case.
A: "Collection" usually means either "set" or "class."
There are two reasons to prefer the word "collection," both of which may apply to the quoted text.  The first is what Daniel said, which is to avoid saying "set...of sets."  The second is to avoid the distinction between sets and proper classes, which is usually not relevant outside of axiomatic set theory (e.g. here the collection is contained in the power set of the topological space, so it cannot be a proper class.)
As Stefan pointed out in his answer, here we need to assume that all collections are nonempty.  However, I don't think is is standard to do so.
