Difference between each, every, any etc.?

Question

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, suppose that $A$ is a closed subset of a metric space, and $B_1,B_2,B_3,...$ are all compact sets, and 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".

Does "its intersection with every compact set" refer to $$A\cap B_1 \cap B_2\cap B_3 \cap \cdots$$ or $$A\cap B_i \:\text{ for each }i?$$

Answer 1

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.

Answer 2

The real issue here is hanging quantifiers (ambiguity due to mixed quantifiers not all specified at the beginning of the sentence), rather than 'each/every/any' per se.

Forced to decipher the ambiguity, I'd say that

• “$A$'s intersection with every compact set” strongly suggests $A\cap B_1 \cap B_2\cap B_3 \cap \cdots,$
• whereas “$A$'s intersection with each compact set” suggests $\forall i\,A\cap B_i.$