What is the difference of the following two mathematical sentences? (Walter Rudin's Principles of Mathematical Analysis)

Question

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 $q \in E$.

Answer 1

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