Is there a reason the word "collection" is used in Heine-Borel theorem to describe an open covering instead of "set"?

The Heine-Borel theorem is stated as follows:

Suppose $\mathcal{H}$ is an open covering of a compact set $S \subseteq \mathbb{R}$. Then $S$ is of an open covering $\tilde{H}$ consisting of finitely many open sets belonging to $\mathcal{H}$.

Here, an open covering $\mathcal{H}$ of $S$ is defined as a collection of open sets such that every point $s \in S$ can be found in a set $H \in \mathcal{H}$. I am wondering if there is a specific reason why $\mathcal{H}$ is called a collection instead of a set. If this $\mathcal{H}$ cannot be constructed using rules from the set theory, how to interpret the use of it?

Solution 1:

This is just for clarity. One of the common stumbling blocks in interpreting a piece of mathematics is the issue of type: keeping track e.g. of what's a set of points versus what's a set of sets of points, and so on. It seems to be the case that using varied terminology often makes things a bit easier to read.

As to formal content, there is none: there are no set theoretic issues with constructing sets of open sets.