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This passage is a quotation from a mathematics book, Analysis on Manifolds by James R. Munkres. My question is about the English usage. Why is compact support in the singular form here?

Roughly speaking, this statement says that if the lemma holds locally for đť‘” and functions đť‘“ having compact support, then it holds for đť‘” and all đť‘“.

In the sentence immediately before this one, the book says:

all continuous functions 𝑓: 𝑉 → ℝ whose supports are compact subsets of 𝑉.

Here, supports appears in the plural. Why is there this difference in usage?

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"Support" in the sense of assistance/help is generally considered to be an uncountable noun. There is no plural form — similar to nouns like advice, information, evidence, milk, etc. However, it is also a countable noun in contexts which mean physical supports, such as mechanical devices used to prop something up. So there is a plural form, depending on usage.

However, in the second example the context here is regarding a mathematical concept, which I have been informed (via comments) is a different usage here, and it is acceptable in maths terminology. I can't verify this, as I am no mathematician.

Note that while it is generally possible to pluralise uncountable nouns in English to talk about "different kinds/types" of something, in many cases using a plural form of an uncountable noun is something that some will avoid as it can sometimes sound quite awkward. Instead of pluralising, we would rather say something like "types of support" or "kinds of milk", "pieces of evidence", etc.

Ultimately you would have to ask the author why they decided to use the plural form here. Not being a mathematician, I can't be 100 percent sure how common this is in maths (and can only go by the comments here), however it wouldn't surprise me at all. In specialist fields of study, the use of English words can often differ significantly from their everyday meaning. Such fields often have their own turns of phrase, jargon, or lingo if you will.

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    This is incorrect: “supports” here is referring to different kinds of support for each function. In mathematics, each function can have only one compact support. Having support with functions is clearly a participle phrase (present participle + singular) modifying the plural noun functions. Unlike uncountable nouns such as advice , support is not the same, and the comparison to “kinds of milk” is misleading. Support and supports reflect the same concept Commented Sep 13 at 17:24
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    Another math person here. Yes, the comment by @JamesMathai is right - "support" in the mathematical sense is not an uncountable noun, and the use of the singular is due to the grammatical role played by the word. Commented Sep 14 at 19:23
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    See edits. Further to comments here I have removed the claim that I made about mathematical "support" being uncountable. I stand corrected. However in general English usage (not in maths) "support" is indeed uncountable in the sense of assistance/help. That I shall not remove, as it is absolutely correct. Commented Sep 15 at 14:19
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I agree that with proper English knowledge, you can understand mathematics better.

There is a well-known saying:

  • Those who live in glass houses shouldn't throw stones at others..

It can be rewritten using "a house," but the meaning remains the same:

  • People living in a glass house shouldn't throw stones at others..

Another sentence:

  • The percentage of students owning a car in our college is 36%.

I want to clarify that the original saying is not normally used in this way. I modified the first sentence only to illustrate the grammar. While it could be read in two different ways, we should consider the logical interpretation. We are speaking about people who live in their own glass houses. It does not mean that a single glass house is crowded with people — just as, in the other example, many students cannot all be owning a single car. In the first sentence, the participle phrase "living in a glass" house modifies people, and in the second, "owning a car" modifies "students."

Similarly, in OP's sentence:
The present participle phrase "having compact support" is a reduced relative clause, functioning adjectivally to modify the noun "functions.".

In mathematics, each function has one support set, and some of these may be compact. So to describe such functions we can use the participle phrase "having compact support", while the plural "supports" is used when referring to many functions together.

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    @tchappyha I hope you read my answer fully and noticed that it takes a completely different direction. I wrote it because I completely disagreed with the explanation given. Commented Sep 14 at 2:12
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    The "owning a car" is a good parallel: it uses the singular even though these students will own different cars. Commented Sep 15 at 8:26
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    Your suggestion that "people living in a glass house" can be used and means the same as the plural is factually incorrect. This is a fixed expression and idiom. It is always "people in glass houses" and is absolutely never said in the singular. check the ngram search Commented Sep 15 at 14:51
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    Also, your claim about participle phrases such as "having support" being a factor here does not matter one jot. If this were a normal English usage of "support" in the sense of assistance/help (not in the mathematical sense) then in such instances "supports" would be incorrect. It is not idiomatic at all to pluralise support in this sense. We would not say something like "I would like to thank our benefactors, whose supports are greatly appreciated". This is really bad/awkward English, as bad as the plural form"informations". This is a common mistake that non-native English speakers make. Commented Sep 15 at 15:21
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    @BillyKerr It’s a well-known saying, and I used it only to illustrate the grammar and meaning. I didn’t mean to suggest that the saying itself is used in this way. I recall someone once pointing out that “people living in a glass house” could be read literally as many people crowded into one house, so I included the sentence to show how it should be logically interpreted. To be on the safe side, I also gave a second example. Commented Sep 15 at 15:27
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To my ears (as a native English speaker and mathematics student of many years),

all continuous functions f whose supports are compact subsets of V

and

all continuous functions f whose support is a compact subset of V

are both grammatical and sound fine. However, the mental picture the first sentence gives is a set V, and lots of functions f compactly supported in V.

Whereas the mental picture of the 2nd sentence is one function compactly supported in V.

If I were the author, I would use the 2nd phrase if I was making an argument where I fix and consider a single function f. Like if I were teaching, and I draw one function f to talk about on the board, then I would say the 2nd phrase; whereas if I drew many functions to illustrate my point, then I would say the 1st phrase.

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  • What about the OP's 1st example: "and functions đť‘“ having compact support" - does that sound natural? Commented Sep 20 at 8:44
  • @BillyKerr yes, sounds natural to me. I parse it as “holds locally for g [and all] functions f having compact support”. Perhaps it would be slightly better to say “functions f who have compact support”, but the difference is negligible. Commented Sep 20 at 21:08

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