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Let's work with open intervals of the real line, $\mathbb{R}^1$. That is, let us deal with objects of the form $(a,b), a,b\in \mathbb{R}$

$\{(a,b): a < b, a,b\in \mathbb{R}\}$ I believe represents the set of all open intervals (please correct me if I'm wrong).

I am wondering how one would denote the family of all open intervals on the real line? Going one step further, how would one denote the family of all countable collections of open intervals on the real line?

I am wondering because I don't see how I can index through all open intervals (I would need to do so to represent a family, wouldn't I...?).

One further question -- I will make a separate topic for it later if needed -- is whether there are general guidelines on how to tell if something is a family, set, collection, or class. That is, I am not asking for the difference between these -- there are plenty of answered questions on that already -- I am asking that if I am given $\{\cdot\}$ with something inside the brackets, how does one know if they are working with a family, class, set, or collection?

Thanks

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One cannot really keep the answer to this separate from the question of what the words "set", "class", "family", etc mean -- because the question rests at least partially on a misconception. It looks like you think these are different things such that it makes sense to ask whether the thing you're looking at is one or the other. But that is often not the case -- quite often the answer will be, "why, it is both, but right now I'm thinking of it as a such-and-such".

Thus:

As a general rule, whenever you see curly brackets $\{\cdots\}$ you can safely assume that they denote a set, unless the surrounding text clearly tells you that something else is going on. This is the most common kind of collection we use in mathematics, and they're not called "set brackets" for nothing.

Each of the other words can be used as just a colorful synonym for "set" -- for example, if you're doing group theory and meet something called a "congruence class", the word "class" tells you nothing whatsoever that "set" wouldn't have told you. It is simply traditional to call those particular sets "congruence classes", but that doesn't stop them from, formally, just being sets. In this case there's nothing to it but a tradition of using particular terminology, which you have to absorb together with the rest of the terminology in that field as you learn it.

Sometimes the words do mean something special, though.

The exception you meet most often in ordinary mathematics is family. This word is sometimes used for wording theorems where one would like to think of something as a set, but where one also wants to be able to handle the case that two of the elements of the something are the same. A set cannot do that, so in order to make the formalism work, we assume that there is an index set and that what we have really is a function from the index set to the elements we're actually interested in -- but that's all just to make the formalism come out right.

In particular, if you want to apply a theorem that demands a family of whatchamacallits, and what you actually have is merely a set of whatchamacallits, that's no problem at all -- you can just declare your set to be indexed by itself; ta-dah, then it's a family. And this is so common that it often isn't even said in so many words; the reader of the argument is supposed to be able to imagine how it goes without explicit prompting!

Thus, if you see

Now apply Theorem 117 [which whats a family of gadgets] to $\{x\in\mathsf{Gadgets} \mid \varphi(x) \}$ ...

it doesn't really make sense to ask whether the $\{\cdots\}$ are a family or a set. It's a set that's being used as a family!

Class also has a particular meaning within axiomatic set theory -- namely something that can contain elements but is not necessarily a set (so you may not be able to put it into a set). In that case, every set is a class -- so again it's not an either/or! -- but not all classes are set. One sometimes sees $\{\cdot\mid\cdots\}$ used as a notation for classes-that-are-not-sets, but generally one can expect some explicit warning from the surrounding text when that is the case.

Finally collection is most often used as a fuzzy informal word meaning "something (not necessarily even a "thing" in whatever formal theory I'm thinking about) that contains particular things, but I don't want to care about whether or not it is actually a set right now". Sets are definitely collections, but one can speak of collections that turn out not to be sets.

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  • $\begingroup$ What you call a family here, I would call a multifamily, or an indexed family. For me, "family" is a synonym for "set". $\endgroup$ – Carl Mummert Nov 23 '15 at 0:54
  • $\begingroup$ Thank you for the thorough explanation. After reading it I have a better understanding. However, I am still wondering: A real analysis book I'm looking at says the family of all open intervals on the real line is a sequential covering class (of $\mathbb{R}^1$. So they are using family instead of "set of open intervals" because the same open interval may be needed to cover different subsets of the real line? Makes sense. Also, let $\Gamma$ be the family of all countable collections of open intervals on the real line. So an element of this is a collection? I still can't picture an element. $\endgroup$ – majmun Nov 23 '15 at 1:04
  • $\begingroup$ @Carl: That is certainly also a possibility. In that part of the answer, I was speaking about what happens in contexts where "family" is not just a synonym for "set". $\endgroup$ – hmakholm left over Monica Nov 23 '15 at 1:04
  • $\begingroup$ @majmun: I think both of those examples must be one of the cases where "family" is -- like Carl uses it -- simply a flavorful synonym for "set", but doesn't have any formal content beyond that. $\endgroup$ – hmakholm left over Monica Nov 23 '15 at 1:06
  • $\begingroup$ @HenningMakholm going through the section again, I think that indeed fits. Sorry that I did not see it earlier, and thanks for the help. $\endgroup$ – majmun Nov 23 '15 at 1:12
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Terminology

In ordinary mathematics (outside formal set theory), it is only convention that determines whether something is called a "family, set, collection, or class".

One reason for the varied terminology is that we often begin with a collection of basic objects (e.g., real numbers). Then we form "sets" of these basic objects (e.g. sets of real numbers, such as intervals). As things get more advanced, we begin to look at sets of sets of basic objects. But that terminology may seem confusing, and it may be convenient to know that "set" in this context always means "set of our initial objects". In that case, authors use words such as "class" or "family" for a set of sets of initial objects.

For example, in the book Measure Theory by Paul Halmos, the term "set" means a set of points, and a set of sets is called a "class". Thus Halmos states that a topology is a set of points $X$ and a class $T$ of subsets of $X$ with particular properties. This convention means that, whenever he writes "set", you know it is a set of points.

In formal set theory, everything is a set, so there is no reason to try this sort of linguistic distinction. But in other areas, it is more common to think of objects, sets of objects, and classes of sets of objects as different "types" of things, which deserve different words.

In formal set theory, the word "class" is used for something else entirely -- there is a notion of a "proper class" which is a kind of collection of objects that is not a set.

Notation

Here is one way to define the set (or class) of all open intervals: $$ C = \{ (a,b) : a,b \in \mathbb{R}, a < b\} $$ Or, you could just say "Let $C$ be the class or all open intervals of reals." There is no special notation, and there is no need to index the intervals just to make a set containing all of them.

As for the family of all countable collections of intervals, you could just call it "$F$".

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  • $\begingroup$ Thank you very much. You answer together with Henning's answer helped me a lot. $\endgroup$ – majmun Nov 23 '15 at 1:12

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