Let $X$ be a set and $S\subseteq\mathcal{P}(X)$ be a family of subsets of $X$. I'd like to know if I can replace this commonly used notation $$\bigcup_{Y\in S}Y$$ by this one $$\bigcup S$$ and be understood by any mathematician outside of set theory; that is, I want to know if the second notation is as well-known as the first one. I saw it used in a famous set theory book so my guess is it's well spread among set theorists. However every other mathematician I've seen uses the first notation.

(The requirement "any mathematician outside of set theory" is naturally an wishful exaggeration. I'll be satisfied if more than 90% of mathematicians outside of set theory understand me. Damn it, I'll pay if 90% of just topologists and analysts understand the notation.)

  • $\begingroup$ I'd say $\bigcup S=\{x\mid \exists z\in S, x\in z\}$ is well-known. $\endgroup$ Sep 27 at 22:46
  • $\begingroup$ I suppose you can also use the second notation, but define it the first time you use it. $\endgroup$
    – Joe
    Sep 27 at 22:46
  • 2
    $\begingroup$ I view $\bigcup_{Y \in S} Y$ as a shorthand for $\bigcup\{Y : Y \in S\} = \bigcup S$. So the second notation is clearer and doesn't need any explanation unless you think your readership needs it. $\endgroup$
    – Rob Arthan
    Sep 27 at 23:02
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    $\begingroup$ The trouble is you use the same typeface for $Y$ and $S$. Most disciplines start with elements that are not thought of as sets (usually lower case letters such as $i$ or $x$ for things like points in geometry), then form set of such elements (usually upper case letters $X$ etc), and family of sets (curly letters \mathcal or \mathscr). $\bigcup\mathscr{F}$ is quite common, but $\bigcup F$ is not. $\endgroup$ Sep 28 at 3:58

3 Answers 3


Disclaimer: awkwardly for this question, I'm not a non-set-theorist.

While standard notation in set theory, my experience is in tension with aschepler's answer: I believe that "$\bigcup S$" is not generally understood outside of set theory.

However, you specify that you are most interested in audiences of topologists. Here things are better in my experience, partly because there is a tighter connection between topology and set theory than between (say) number theory and set theory and partly because topologists more frequently care about multiple "types" (= points, sets, sets-of-sets, etc.) of object at once in a way that makes set-theoretic notation more convenient. Ultimately, though, even here (with the exception of general topology) I would not assume that "$\bigcup S$" is guaranteed to be recognized; at the very least, I would suggest reminding your readers of the meaning of the notation when it is first used.

But really, "$\bigcup_{Y\in S}Y$" isn't much longer to write, so given that it's much more common outside of set theory why not use it?


Yes, this is standard in set theory. For example, where the Wikipedia page on unions first goes beyond the binary unions $A \cup B$, it gives the definition

$$ x \in \bigcup \mathbf{M} \iff \exists A \in \mathbf{M}, x \in A $$

and then goes into how the $\bigcup$ symbol can also be used with subscripting.

I don't see these as alternative styles where one would want to consistently use one or the other. The notation which is more convenient depends on what's already defined or to be used: If $S$ contains the sets to be combined, use $\bigcup S$. If $A_i$ are the sets to be combined, use $\bigcup_{i=1}^N A_i$ or similar.

A short note about the meaning of $\bigcup$ without a subscript might be good if the context or audience isn't in set theory.


I have a 4 year undergrad maths degree and have been recently self-studying Set theory. I believe that I have never seen the notation $\bigcup S$ outside of set theory (although I never studied topology). I initially found it a bit confusing, but it doesnt take long to get used to. Inside set theory $\bigcup S$ is extremely common and works its way into many proofs.

Halmos, Naive Set Theory, section 4 notes:

"... U $=\{x|x \in X$ for some X in $\mathcal{C}\}$. This set U is called the union of the collection $\mathcal{C}$ of sets; … The simplest symbol for U that is in use at all is not very popular in mathematical circles; it is $\bigcup\mathcal{C}$. Most mathematicians prefer something like $\bigcup \{X|X\in\mathcal{C}\}$ or $\bigcup_{X\in\mathcal{C}}X$."

  • $\begingroup$ Very enlightening, thank you. Interestingly enough, my abstract algebra teacher recently used the $\bigcup \mathcal{F}$ notation in his lesson. $\endgroup$ Oct 4 at 14:00
  • $\begingroup$ However I fail to understand what kind of object $\bigcup$ is for the author. Does he consider it to be both a set and an operator? $\endgroup$ Oct 4 at 14:02
  • $\begingroup$ Hi Lucia, I’ve updated my post as I misquoted. The letter U is used for the set, not the operator $\bigcup$. $\endgroup$
    – Porky
    Oct 4 at 21:30
  • $\begingroup$ Thanks. Luca though 😗 $\endgroup$ Oct 4 at 21:32

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