I am very poor at English.
But I am reading Walter Rudin's Principles of Mathematical Analysis now.
In the book, I found the following sentence:
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q \neq p$ such that $q \in E$.
I cannot understand why "the set $E$" instead of "a set $E$".
What is the difference in meaning between the following two mathematical sentences?
(1)
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q \neq p$ such that $q \in E$.
(2)
A point $p$ is a limit point of a set $E$ if every neighborhood of $p$ contains a point $q \neq p$ such that $q \in E$.
There's no fundamental difference; to say "the set $E$" or "a set $E$" are both equally valid, grammatically, and Rudin isn't referring to a special or particular set either. It's just how some people word things sometimes.
While not particularly mathematical, this did remind me of another Stack Exchange question I saw recently. You might find it worth looking at: here ya go