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In topology, there is a very strong need of describing subsets of some product $X\times Y$ by means of unions and products.

For example, this is a very convenient way of describing subsets of the plane. (Which are common examples in, say, algebraic topology.)

Is there any standard convention about the precedence of products and unions?

In particular, when I write $A\times B\cup C\times D$ would this be interpreted as $A\times(B\cup C)\times D$ or as $(A\times B)\cup(C\times D)$? I personally would go with the second option, because of the analogy with multiplication and addition, which for some reason seems to make sense to me. (And in that case there is a standard convention). Despite this analogy, I fear the omission of parentheses in the case of sets might be ambiguous to some people.

Is it better to always use parentheses?

(Of course a downside of parentheses would be that the notation may become very cluttered, when the expressions are long.)

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I would go with the first interpretation since $B \cup C$ is one object, not two disparate ones. This is, of course, unless I know/understand the context. When there's chance of confusion, use parentheses. See etc... – Tyler Jan 21 '13 at 16:51
+1, nice question! Personal opinion: second, alike in multiplication and addition. I would also say that especially in long formulas like that I really do like parentheses, as without them it's much easier to get it in an incorrect way. – Ilya Jan 21 '13 at 16:52
@Ilya: A very good point. My fear of notation becoming cluttered just might be unjustified. – Dejan Govc Jan 21 '13 at 18:07
Opinion: Third choice: always use parentheses. – GEdgar Jan 21 '13 at 18:09
I agree very strongly with @GEdgar: always use parentheses. – Brian M. Scott Jan 21 '13 at 19:05

If the context is product topology, I think it is more likely to mean the latter as A and C would often be open sets of some space X and B and D would be open sets of some space Y. And it doesn't make sense to take the union of B and C in that context.

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