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(Disclaimer: I've been debating with myself whether to post this question here or on English Learners SE or on Mathematics Education SE, but decided to post here. If the community thinks it's inappropriate, I'll happily agree to migrate the question.)

My question is not about the math per se, but about terminology. Is it better to say

Let $f$ be a function defined on a closed and bounded set $R$

or

Let $f$ be a function defined on a closed bounded set $R$

(In the context of a typical Calculus textbook we can't use the word "compact".) As minor as it seems, I think there's a difference, and I'm very much in favor of the latter one. I've seen a Calculus textbook that consistently uses closed and bounded, which seems okay... until the phrase

(certain property) is true for closed and bounded sets.

To me, this gives an impression that the said property is true for two types of sets: for closed sets and for bounded sets. But that impression is false, because what they mean to say is the simultaneously closed bounded (i.e. compact) sets.

I've been asked to give my feedback on this text. Should I suggest changing closed and bounded into closed bounded, or is it okay as it is?

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    Don't use the latter one without a comma, at least. "A closed, bounded set $R$". "Closed bounded" is not a single adjectival phrase, after all. – 2017-01-29
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    @Arthur: Yes, I know what they mean. My question is: if we need to choose the best and mathematically most accurate way to express this in English, which one is it: `closed and bounded` or `closed bounded`? Or are both of them equally acceptable? Or is one definitely better than the other? – 2017-01-29
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    "Closed and bounded" is safest. But there is no big difference in my opinion. – 2017-01-29
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    @Arthur: Thank you! I'm still not convinced (except for the comma correction -- I didn't think of that, but I get it now!), but I appreciate your opinion. – 2017-01-29
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    Usually one knows from the context what is meant. One solution is to write "closed-and-bounded" (using dashes, or hyphens). (I have seen similar use for "closed-and-discrete".) Another example for abuse of language is "countable union" which usually means union of countably many sets (the union of a countable family of sets), though one may argue that it should mean the union of any family of sets (a countable family or not), as long as the result of the union is a countable set. – 2017-05-04
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    @Mirko: True, but _"usually"_ doesn't apply to textbooks, where the authors have to be more careful exactly because textbooks are addressed to students who don't know the subject yet. – 2017-05-05

1 Answers 1

2

(A) The phrase

(certain property) is true for closed sets as well as for bounded sets.

indeed informs us that the said property is true for two types of sets: for closed sets and for bounded sets.

(B) On the other hand, the phrase

(certain property) is true for closed and bounded sets.

might occasionally give a false impression similar to (A); and this false impression is more likely when the reader/listener is a non-native English speaker (such as myself, years ago). Nevertheless, (B) actually tells us, in proper English, that the said property is true for sets that are simultaneously closed and bounded.

Addendum

(C) The phrase (without commas)

(certain property) is true for closed or bounded sets.

conveys the same meaning as (A). However, if someone misreads (C) as

(D)

(certain property) is true for closed, or bounded, sets. $ \ \ $ (note the commas)

then the meaning would become entirely different and very misleading: the property is true for closed sets, which is another name of bounded sets.