# 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?

• set, collection, family --- these are synonyms in the literature, simply for the sake of linguistic variation. Jan 6, 2021 at 21:09
• To add to what @Ittay Weiss said, sometimes in non-set theoretic applications "collection" and "family" and other such words are used for sets of sets (e.g. a collection of open sets) or for sets of sets of sets (e.g. a family of open covers of a set). Jan 6, 2021 at 21:15

• Also stumbling is language: "Let $X$ be a set of sets and let $x$ be a set in the set $X$. Suppose the set is countable ...", which is "the set"? When one is a set and one is a family and another is a collection it's easier for the reader to keep track of what's going on. It's a compression scheme where we make information more accessible by spreading it over different words rather than more words. Jan 6, 2021 at 22:45