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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?

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    $\begingroup$ set, collection, family --- these are synonyms in the literature, simply for the sake of linguistic variation. $\endgroup$ Jan 6, 2021 at 21:09
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    $\begingroup$ 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). $\endgroup$ Jan 6, 2021 at 21:15

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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.

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    $\begingroup$ 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. $\endgroup$
    – Asaf Karagila
    Jan 6, 2021 at 22:45

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