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I'm having a bit of trouble understanding the use of these different phrases. Are there any certain rules for when which of the word is to be used? I feel like they overlap a lot, but can bear slightly different meanings.

For instance, look at this task from my text book: "Prove that in a metric space a subset is closed if and only if its intersection with every compact set is closed".

Suppose $A$ is a closed subset of a metric space, and $B_1,B_2,B_3,...$ are all compact sets. Does "its intersection with every compact set" refer to $A\cap B_1 \cap B_2\cap B_3 \cap \cdots$ or just $A\cap B_i$ for each $i$?

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    For the specific problem, it means the latter. The general question is more an English question. The usage in mathematics should be the same as in everyday language, with the possible exception that "any" is hardly ever used in mathematics as "pick an arbitrary one you like." – 2017-02-26
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    Thanks! Perhaps a bit off topic, but I think the word "every" suits the first alternative the best, whereas "each" would suit the latter. Though, I realize now the latter is the one it's supposed to be in this problem as there will always (?) exist a $B_i,B_j$ such that $B_i\cap B_j=\emptyset$. – 2017-02-26
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    I agree that "each" would be somewhat more clear, though I might not be optimally placed to tell. But it is I think not true that every would fit the former. I will elaborate in an answer. – 2017-02-26

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For the specific example it ought to be meant that the intersection of $A$ is taken with one compact set at a time. (The other option leads to a very false statement.)

I think both usage of 'every' and 'each' can be considered as correct according to the rules of English grammar. Personally, I'd strongly prefer 'each' to avoid the confusion that you highlight, but I am not a native speaker.

The general usage in mathematics follows by and large the general rules of English grammar. From a grammar site:

The difference between All, Every, and Each - Quick Explanation

  • All means the total number of people or things considered as a group.
  • Every means all members of a group considered individually.
  • Each means all members of a group considered individually though we think of them more one by one.

Thus both 'either' and 'each' mean that members of a group are considered individually, the 'each' just stresses it.

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    As a native English speaker, I find the text cited by the OP to be unclear and I am very sympathetic with the OP's reading of "intersection with every compact set" as meaning the intersection with the intersection of all the compact sets (which is meaningful). What does "$1$ plus every number between $5$ and $97$" mean? Who knows? It's just bad mathematical English. Don't defend the indefensible! – 2017-02-26
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    I changed my post a bit in that direction. "Meaningful" was not a good choice of word. I meant it is so clearly false that it cannot have been the intended meaning. On your example: I agree it is unclear, but honestly I'd not be sure what "1 plus *each* number between 5 and 97" would mean either. – 2017-02-26
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    Exactly: you can't fix "$1$ plus every/each/some/any number between $5$ and $97$". The OP's cited text includes a noun phrase like that, so it needs to be rewritten. (Apologies if I sound harsh as a native English speaking mathematician shouting at my fellows to write better, while you are doing your best to interpret what they have written.) – 2017-02-26